{"id":1545,"date":"2023-10-31T11:26:17","date_gmt":"2023-10-31T11:26:17","guid":{"rendered":"https:\/\/myknowledgehub.org\/?p=1545"},"modified":"2023-12-13T00:28:39","modified_gmt":"2023-12-13T00:28:39","slug":"research-methodology-chapter-7-3-a","status":"publish","type":"post","link":"https:\/\/myknowledgehub.org\/index.php\/2023\/10\/31\/research-methodology-chapter-7-3-a\/","title":{"rendered":"Research Methodology Chapter 10.2"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"1545\" class=\"elementor elementor-1545\">\n\t\t\t\t<div class=\"elementor-element elementor-element-52db2cb e-flex e-con-boxed e-con e-parent\" data-id=\"52db2cb\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t<div class=\"elementor-element elementor-element-7d25bd5 e-con-full e-flex e-con e-child\" data-id=\"7d25bd5\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-1fe6d70 elementor-widget elementor-widget-image\" data-id=\"1fe6d70\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"682\" src=\"https:\/\/myknowledgehub.org\/wp-content\/uploads\/2023\/11\/rat-cell-an-animal-in-a-cage-5088740-1024x682.jpg\" class=\"attachment-large size-large wp-image-1990\" alt=\"rat, cell, an animal in a cage-5088740.jpg\" srcset=\"https:\/\/myknowledgehub.org\/wp-content\/uploads\/2023\/11\/rat-cell-an-animal-in-a-cage-5088740-1024x682.jpg 1024w, https:\/\/myknowledgehub.org\/wp-content\/uploads\/2023\/11\/rat-cell-an-animal-in-a-cage-5088740-300x200.jpg 300w, https:\/\/myknowledgehub.org\/wp-content\/uploads\/2023\/11\/rat-cell-an-animal-in-a-cage-5088740-768x512.jpg 768w, https:\/\/myknowledgehub.org\/wp-content\/uploads\/2023\/11\/rat-cell-an-animal-in-a-cage-5088740.jpg 1280w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-8e9fd94 e-con-full e-flex e-con e-child\" data-id=\"8e9fd94\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-610e480 elementor-widget elementor-widget-heading\" data-id=\"610e480\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Student's t-Test<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-26ed793 e-flex e-con-boxed e-con e-parent\" data-id=\"26ed793\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0976271 elementor-widget elementor-widget-heading\" data-id=\"0976271\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">I. Basic Concepts of Student's t-Test<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-cd1fa87 e-flex e-con-boxed e-con e-parent\" data-id=\"cd1fa87\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-1f77a5e elementor-widget elementor-widget-text-editor\" data-id=\"1f77a5e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Student&#8217;s t-test is a statistical test that is used to determine if there is a significant difference between the means of two groups. It is a commonly used test when the sample size is small and the population standard deviation is unknown. The t-test allows researchers to make inferences about the population based on the sample data.<\/p><p><strong><em>Student&#8217;s T-test is a statistical test used to compare the means of two groups of data<\/em><\/strong>. It is a parametric test, which means that it assumes that the data is normally distributed. The test compares the means of two groups and determines if the\u00a0difference between them is statistically significant. The t-test calculates a\u00a0<strong>t-value<\/strong>, which is then compared to a\u00a0<strong>critical value\u00a0<\/strong>to determine if the difference is significant.<\/p><p>The T-test is used in a variety of fields, including research, medicine, and business. For example, researchers might use the T-test to compare the effectiveness of two different treatments or to compare the average test scores of two different groups of students.<\/p><p>There are three main types of t-tests:<\/p><ul><li><strong>One-sample t-test: <\/strong>This test is used to compare the mean of a sample to a known population mean.<\/li><li><strong>Independent samples t-test: <\/strong>This test is used to compare the means of two independent groups of data, where the groups are not related to each other.<\/li><li><strong>Paired samples t-test: <\/strong>This test is used to compare the means of two related groups of data, where the groups are paired together.<\/li><\/ul><p>\u00a0<\/p><p>When performing a t-test, we compare the calculated t-value to the critical value at a given level of significance. If the calculated t-value is greater than the critical value, we reject the null hypothesis and conclude that there is a significant difference between the means of the two groups. If the calculated t-value is less than the critical value, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference between the means.<\/p><p>It is important to note that the t-test only provides evidence of a difference between the means, but it does not provide information about the direction of the difference. To determine the direction of the difference, additional analysis or interpretation is required.<\/p><p>The Student&#8217;s t-test is a powerful statistical tool that allows researchers to compare the means of two groups and determine if there is a significant difference. By understanding the basic concepts and different types of t-tests, researchers can make informed decisions and draw meaningful conclusions from their data.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-68fa8a6 e-flex e-con-boxed e-con e-parent\" data-id=\"68fa8a6\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-cee20d2 elementor-widget elementor-widget-heading\" data-id=\"cee20d2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">II. Types of Student's t-Test <\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f580e78 e-flex e-con-boxed e-con e-parent\" data-id=\"f580e78\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f06cd90 e-flex e-con-boxed e-con e-parent\" data-id=\"f06cd90\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-df0428f elementor-widget elementor-widget-heading\" data-id=\"df0428f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">A. One-Sample t-Test<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-844a7e1 e-flex e-con-boxed e-con e-parent\" data-id=\"844a7e1\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5b0cd1e elementor-widget elementor-widget-text-editor\" data-id=\"5b0cd1e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The\u00a0<strong>one-sample t-test is used to compare the mean of a sample to a known population mean<\/strong>. For example, a company might use the one-sample t-test to compare the average weight of its employees to the national average weight, or a researcher might use the one-sample t-test to compare the average IQ of a group of students to the general population.<\/p><p data-sourcepos=\"79:1-79:219\">A company wants to know if the average weight of its employees is different from 170 pounds. The company collects a sample of 100 employees and finds that the average weight of the employees in the sample is 165 pounds.<\/p><p data-sourcepos=\"81:1-81:81\">To perform a one-sample t-test, the company first needs to calculate the t-score:<\/p><p data-sourcepos=\"81:1-81:81\"><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"81:1-81:81\"><span style=\"font-style: inherit; font-weight: inherit;\">where:<\/span><\/p><ul><li>x\u0304 is the mean of the sample<\/li><li>\u03bc is the hypothesized population mean<\/li><li>s is the standard deviation of the sample<\/li><li>n is the sample size<\/li><\/ul><p data-sourcepos=\"94:1-94:29\">\u00a0<\/p><p>In this case, the t-score is:<\/p><p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"94:1-94:29\"><span style=\"font-style: inherit; font-weight: inherit;\">\u00a0 \u00a0 \u00a0t = &#8211; 2.5<\/span><\/p><p data-sourcepos=\"94:1-94:29\"><span style=\"font-style: inherit; font-weight: inherit;\">The company then needs to look up the p-value for a t-score of -2.5. At degrees of freedom of 99 and a level of significance of 0.05, the critical t-value is \u00b11.984.\u00a0<\/span><\/p><p data-sourcepos=\"104:1-104:67\">Since the calculated t-score (-2.5) is outside \u00b11.984 (or since the calculated t-score (-2.5) is greater than the critical t-value (1.984))<span style=\"font-style: inherit; font-weight: inherit;\">, we would reject the null hypothesis.\u00a0<\/span><\/p><p>This means there is enough evidence at the 0.05 significance level to conclude that the average weight of the employees is different from 170 pounds. The negative sign of the t-statistic indicates that the average weight is less than 170 pounds.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-dabb591 e-flex e-con-boxed e-con e-parent\" data-id=\"dabb591\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-6b7e507 elementor-widget elementor-widget-heading\" data-id=\"6b7e507\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">B. Independent Samples t-Test<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-3cdb62e e-flex e-con-boxed e-con e-parent\" data-id=\"3cdb62e\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-b787b67 elementor-widget elementor-widget-text-editor\" data-id=\"b787b67\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The\u00a0<strong><em>independent samples t-test is used to compare the means of two independent groups of data<\/em><\/strong>, where the groups are not related to each other. For example, a researcher might use the independent samples t-test to compare the average height of males and females or to compare the average test scores of students from two different schools.<\/p><p data-sourcepos=\"14:1-14:339\"><strong>Example<\/strong><\/p><p data-sourcepos=\"15:1-15:224\">Researchers want to compare the average height of males and females. They collect a sample of 100 males and 100 females and find that the average height of the males is 175 cm \u00b1 10 cm and the average height of the females is 165 cm \u00b1 10 cm.<\/p><p data-sourcepos=\"17:1-17:94\">To perform an independent samples t-test, the researchers first need to calculate the t-score:<\/p><p data-sourcepos=\"17:1-17:94\"><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"17:1-17:94\"><span style=\"font-style: inherit; font-weight: inherit;\">where:<\/span><\/p><ul><li>x\u0304<sub>1\u00a0<\/sub>and<sub>\u00a0\u00a0<\/sub>x\u0304<sub>2 <\/sub>are the means of the two groups<\/li><li>S<sub>1<\/sub> and S<sub>2<\/sub> are the standard deviations of the two groups<\/li><li>n<sub>1<\/sub> and n<sub>2<\/sub> are the sample sizes of the two groups<\/li><\/ul><p>In this case, the t-score is:<\/p><p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"29:1-29:29\"><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\"><br \/><\/span><\/p><p data-sourcepos=\"29:1-29:29\"><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">That simplified will be\u00a0<\/span><\/p><p>t = 7.072<\/p><p>The researcher then needs to look up the t-crtical value for 198 degrees of freedom (df) at 0.05 level of significance. The t-critical value is found to be 1.975. The t-score is greater than the t-critical value, hence we reject the null hypothesis.\u00a0<\/p><p data-sourcepos=\"41:1-41:220\">Alternatively, we can look up the p-value for a t-score of 7.072 with 198 df and take a decision. The p-value is the probability of obtaining a t-score as extreme or more extreme than the one they observed, assuming that the null hypothesis is true. The p-value for a t-score of 7.072 with 198 df is less than 0.0001. This means that there is a very small chance of obtaining a t-score as extreme or more extreme than the one they observed if the null hypothesis is true.<\/p><p data-sourcepos=\"43:1-43:149\">Therefore, the researchers reject the null hypothesis and conclude that there is a significant difference in the average height of males and females.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-d0c02b6 e-flex e-con-boxed e-con e-parent\" data-id=\"d0c02b6\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5bd34f4 elementor-widget elementor-widget-heading\" data-id=\"5bd34f4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Two-Sample t-Test Assuming Equal Variances<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-2d1524b e-flex e-con-boxed e-con e-parent\" data-id=\"2d1524b\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0bf14e1 elementor-widget elementor-widget-text-editor\" data-id=\"0bf14e1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The two-sample t-test assuming equal variances is used when the variances of the two groups are assumed to be equal. This assumption is often made when the sample sizes of the two groups are similar and the data is normally distributed. For example, if we are comparing the test scores of students who received different teaching methods, we can use the two-sample t-test assuming equal variances to determine if there is a significant difference in the mean test scores.<\/p><p>To perform this t-test, we calculate the mean and standard deviation of each group. The pooled standard deviation is then calculated by combining the standard deviations of the two groups. The t-value is calculated by dividing the difference in means by the standard error of the difference. This t-value is then compared to the critical value to determine if the difference is statistically significant.<\/p><p>The formula for calculating the t-statistic in this case is as follows:<\/p><p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"17:1-17:94\"><span style=\"font-style: inherit; font-weight: inherit;\">where:<\/span><\/p><ul><li>x\u0304<sub>1\u00a0<\/sub>and<sub>\u00a0\u00a0<\/sub>x\u0304<sub>2<\/sub> are the means of the two groups<\/li><li>S<sub>p <\/sub>is the pooled standard deviation (see below for calculation)\u00a0<\/li><li><span style=\"font-style: inherit; font-weight: inherit;\">n<sub>1<\/sub> and n<sub>2 <\/sub>are the sample sizes of the two groups<\/span><\/li><\/ul><p>The Pooled Standard Deviation is calculated using the following formula:<\/p><p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><ul><li><span style=\"font-style: inherit; font-weight: inherit;\">S<sub>1<\/sub> and S<sub>2<\/sub> are the standard deviations of the two groups<\/span><\/li><li><span style=\"font-style: inherit; font-weight: inherit;\">n<sub>1<\/sub> and n<sub>2<\/sub> are the sample sizes of the two groups<\/span><\/li><\/ul><p data-sourcepos=\"5:1-5:9\"><strong>\u00a0<\/strong><\/p><p data-sourcepos=\"5:1-5:9\"><strong>Example:<\/strong><\/p><p data-sourcepos=\"5:1-5:9\"><span style=\"font-style: inherit; font-weight: inherit;\">Suppose we want to compare the average scores of students from two different tutoring programs. We have a sample of 23 students from Program A and another sample of 30 students from Program B. The average score for Program A is 75 with a standard deviation of 8, and the average score for Program B is 72 with a standard deviation of 7. Assume equal variances.\u00a0<\/span><\/p><p data-sourcepos=\"5:1-5:9\"><span style=\"font-style: inherit; font-weight: inherit;\">We know that for Program A, n =\u00a0 25, sample mean (x\u0304<sub>1<\/sub>) is 75, and standard deviation (s1) is 8, while for Program B, n = 30, sample mean (x\u0304<sub>2<\/sub>) is 72 and standard deviation (s2) is 7.\u00a0<\/span><\/p><p data-sourcepos=\"5:1-5:9\"><span style=\"font-style: inherit; font-weight: inherit;\">First, we calculate the Pooled Standard Deviation using the formula:<\/span><\/p><p data-sourcepos=\"5:1-5:9\"><img decoding=\"async\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWsAAACDCAMAAAB8+MoRAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAMAUExURf\/\/\/wAAAN7W1s7OzsW9xa2tte\/v7xAQEDEZKTo6ShAQKaVajEJajHNajBBajBBCWq2MnK1aWqUpWqVaKWspWmtaKYxaWqUIWqVaCGsIWmtaCKWlGULvGaUZrUIZraUp3kIp3u+cpXPvWhCtWnPvGRCtGXMZrRAZrXOlWnvvnBDvWnOlGRDvGXMp3hAp3nvv3nuU7973tUrO3kLOWqWEWkrOnKWEGULOGaUI3kII3s6cpRDOWnOEGRDOGXMI3hAI3joQWhAQWjExMebe5oyMlGNjUqVa70Ja76VarUJarXNarRBarXNa7xBa7xBjWqVazkJaznNazhBazq3v3q2U7xBjGe8Qpe8ZEM4Qpc4ZEHOElGtra1pSWs7W3ubOtebOlDpjWu8Q5u9aEO\/OEO+UEBmM7xmMrc4Q5s5aEM7OEM6UEBmMzhmMjK2tpZScnO8xpe8ZMe9zpe8pY85zpc4pY84xpc4ZMe9Spe8IY85Spc4IY7W9tVpKSjoQCN73OqXOa0qM70KMa0qMraXOKUKMKaUpjEIpjHPOaxCMa3POKRCMKXMpjBApjHvOrXvO73ulzt73EN73Y6XOSkqMzkKMSkqMjKXOCEKMCKUIjEIIjHPOShCMSnPOCBCMCHMIjBAIjHvOjHvOznuEznOEUnN7cwAICO\/Oc+9z5u9ac+8x5u9aMe\/OMaUpKe+15u+UMe+Uc2spKRmt7xmtrRnv7xnvrc7Oc85z5s5ac6UIKc615s6Uc2sIKRnO7xnOre\/OUu9S5u9aUs4x5s5aMc7OMaUpCO+U5s6UMe+UUmspCBmtzhmtjBnvzhnvjM7OUs5S5s5aUqUICM6U5s6UUmsICBnOzhnOjHOlnK3vrTpjKa3O762lzhBCKTo6EK3FlK3vjDpjCK3Ozq2EzhBCCKXva0rv70Lva0qt76Wla0Kta973hEqtraXvKUKtKUrvraXvSkrvzkLvSkqtzqWlSkKtSkqtjKXvCEKtCErvjMXOpdb35u\/W1t73\/8XWxQAAEP\/v\/wAAADzF2eAAAAEAdFJOU\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/wBT9wclAAAACXBIWXMAABcRAAAXEQHKJvM\/AAAJ60lEQVR4Xu2da3LiOBCAkUz0yK+cSYhLjJyquYB9Afk2K+60VftnrmGKmVq9\/MAWxAaMk0l\/YbBIwJbbre5Wq81sAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD49OBf29M209\/ph8RTfzaYom9HGc\/92eCDzL4bq+k1ymILWBqNcGwBSwOyfh5vtI4tYGlKuostYGkkW8srfz+UiA1gcfY8NoDFyfexASyOULGxJERLqf\/EF+uRlbI8xvYaPEPWmvHCUL62E86pLAXT8dUKIBkblvsTYOk9nCr7a4mK+PI62f05A5KUJ9nbDmD0M75cAWRiY3Pi4u7TVKx36c4pJqXXiGL3Z+GI4BevK10v7iKHplclUneoNQkzfVLS\/MJeOJ2w+4yKu8Y4Dscghl645gV6i63ng2mUddEp+A1gLqIk8SEPjQFlc6Br1Owurftj2oOUfdvYgemKYVd2ePFbTO+Is4lCqBVSmbQVMiHqqIQ9KnRPcqZkCP2K7U2eyhXj12dEXZfQLHTJoK3f3oIWOe9kvWEJY1EkRE3ocARodIcosBJ72p0FRlVsddQsPeaeREFPbkN+N7IiXlBzolCiys0JdcPCtIp9fI+qq32kNbgEhA4lm7ei8r0goS8TKa27Eaiz9uIQk8UE4+hMuDvgnH0+lpL5frQaJanVwIwjNsG69ih6eq1R0J6d+bmvGLfjxprhl6KoBqIlQy0jzYggBtnhpgWaGxixnqxlvORvP3MlhGsbWhaF4avl62XwaSa6EsMNyrWQ+56eTuGlJ2t88G3CqR0dyplNmVxSJUM\/peNOSJ4rZAouebxqUyGiZwmjs\/fuw6tSFlaxx6blWZhXL+s8TDSIJDYArU42FGziimk0YvJQ6lQH+6FS7+1+yLHeHeuhKyT\/DSRZxMFFpP0wz7G1TPMEY2Xd6XUW1IUjd1jjTu9Y207s1lsaMUFGvO1khpCNTOy5xtfT6NuQDTu488GIvYfXFxjZkEbWlgJROx7m+krStyFZ6FKeDItWoQoyyp2APYUft+loO6cdh7M1hjO9ZmEJ0yB64TSPwu8CHfymlXhP1tIfXyaDZOM\/1dAJ19LX66guJ3TXHO2R8CCjqu2k8i3VJC90zzsRuVc9+mdwptfBhlhRHS6cZm3c5\/cH5jftBbEz19iyQnPH5cgFSUSXRva6UbhPtfQ9ndXr7n1Rr29xsQvhwyCng40Gsd+u98I\/bzJFp80m+3pdtyqPRXRuW9MXSQMbWIhtu5PgLmxc4l78ZLKopsw6z+11O0rsTMsFW0SqnFt\/tB4ixAJlYzOwP18cMjSS5xNn7v04ZNt5tAw5sWuFUgvIo\/gaUxZbIWwM5pooK2YyyX+cybpngLgbpZxqEgzkWoggyx2KcUfQBneWWhNNyA16XaIf7tkroqD15pSX\/DWl16\/DE+eNCQjm2uU0ivjBaYmSfsxXOQOEffd9qH0KVinVkeeAfZBvyWMnlHfbBeKln+rhS\/myc4hEzM8\/HcLvSjiN9i52R6w7S4xdQv+JrYY2YcS9zCRSJpraSQlZoqnbQVj\/CVH+m1N0wtoAVswLZR9K3eSbNQrGJKiWHYzhJPFoHp1AV8LOEWgVsqkxUDOCK8VNOHETveU5r0O70Ewc7eHd9sSQm3VaJnTjT62qV9sLwYMZ+eG9u+bW\/1Z5owfTcugLgduYKcQfmyZ1EGWDf0+wIbsswxi\/ZyGz0igROZ22jRKpVHEV+Xc0ZpqAetCL96EXTUEybXtxwqEK9RQtM6m1bq\/zdqJFXAbdxaf5eagawHN7h7mbhAxRSb1OkIzrsXCTz1lsU6ubGVtT1LZPnWQMy0faN9Fet0hapebAasqajKdgo4UZwp2I6hniJslQdbbePBgd4ugAHkfBv9ClJa00MjE2LGl7nYSEAKaHjf3L0swphTuaxNja+EvWGZSnU3ondIkiF4jmE2zldXDGkBwvwkwEs4NLz02K+a6hkOAizEfX4e3q4CZWQATfLKUIMfnPqsqH6joZ4okvbobsK26FPW+YjtBG5aa4bR+zhibwg8lMs96EaQ4xpQpMIqxhFskAViqlXErN\/0umXdS8NPU3Jy4\/4FRloksT\/6YUuSeanJ3mPk1kJyIfPU7NU7rRveMS9o\/t28433ZPdjB7hJz7Cm3qfHbzoHldoFpznU7N7cuFhvdVNsb8TN0\/UM4Fur8sKVarZiy703\/8v\/NjW7TE2UeiQFDbRRY\/UW8jrulOpL4j0kb6LQc\/iUMLimPEko7tLK4JAAuziD0Lpn1rmIsOl6OdcSH2sdztXL+Ce4y\/71BNrogEHd6lDQvkGlwSpguCJi1aBHch6BoJKXXDhrL1fpB8XyV0DbvyfQ1bYaUowutop6bj48xq\/emuhX4GsNPLl9gj3cShmbXfWrR5PoAiVk1+EUtAqZ8gP4pXxq6nGl69MJVapLg8pH3Ag6iYhpFq17iDgHKRV61kBsxSpVZQF0LOG2wWCeSxQKPdck4zSsoxlTFOJFcH3svOrHdsrax5j06ZdYhL3S9g+JPT1dJi+yLMUJSqkmWNALPuHpFQLZidKJEdIXPx6DD3IQWQcWbcsKZpUO3bG28wS2iUwKDZm8JCUquaFQqUymF9O6uBzvSZcviEjq8zMN77V+nMCwppiuBnwRxTZ49PmB3L1U92NMiO2538ienNCruijud9jOm3l4YrUIpbmzUE8QtaWEBu0hcRn+JyxRsZvO\/dQ+OLRl1h8c467m6jjzKXMvziLcIObu+HyJAkV1+yQ6EIRU1+B7nihcNAkL0+8vybSHxHZ5xD1LTxI1tnB2f1YSLzZvPfqTP7ULy6hK5Hy2+4P4Za4UDdKSpNXZXehsHtrS+\/+v2z9L2u4lbZy8k6CDY1VpplC46AsG5ryszrviha4d1PBRXoFp1+OY1MRfCfBUBufXMlyWaVkPbiqTZ23H1g+yg43bF0lDyvYvdHx+SEy6FA2P7pNQULFtRUWVoTgzY+xrAdxSGOuXRG6idUW4sM4qkDyaH1lia7fX\/apKAUKLun9MV8Sg5EXE0d5sP+muTu5Y6TXwudtJBIVDxOgj0ujCUOU0gNFXygRXORahfI4TR+iITrU2p5YNEmJSslhPoSE78AgvKkdzT4uItXsP8peBWXPSuI8BuxTZptiLJRHkJD1UV0fQdmqXzaxLEGxC39P28Mxs782dO3S6EUh\/i4e+bHrvwU5ttcf4Eujs0UG2SfAuBoGuYReEyxQOa+GVyFeVeEm3b+RmlnFXqRKdZ9zwWdVW5NcONZPlS6FRBS+t\/ZJ2AmIUV82kfPFKBF9ynepAhY7e\/yLI63PRbHef2rz\/eDDJAWwGL4oDXgOjyhGAgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAeDCbzf+kKrZ9gvLMNAAAAABJRU5ErkJggg==\" alt=\"\" \/><span style=\"font-style: inherit; font-weight: inherit;\"><br \/><\/span><\/p><ul><li><span style=\"font-style: inherit; font-weight: inherit;\">S<sub>1<\/sub> and S<sub>2<\/sub> are the standard deviations of the two groups<\/span><\/li><li><span style=\"font-style: inherit; font-weight: inherit;\">n<sub>1<\/sub> and n<sub>2<\/sub> are the sample sizes of the two groups<\/span><\/li><\/ul><p>By substituting values in the formula, we get<\/p><p data-sourcepos=\"5:1-5:9\"><img decoding=\"async\" style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"5:1-5:9\"><em>S<sub>p <\/sub><\/em>= 7.46<\/p><p data-sourcepos=\"5:1-5:9\"><span style=\"font-style: inherit; font-weight: inherit;\">Now, we calculate the t-statistic using the formula:<\/span><\/p><p>\u00a0<\/p><p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"17:1-17:94\"><span style=\"font-style: inherit; font-weight: inherit;\">where:<\/span><\/p><ul><li>x\u0304<sub>1\u00a0<\/sub>and<sub>\u00a0\u00a0<\/sub>x\u0304<sub>2<\/sub> are the means of the two groups<\/li><li>S<sub>p <\/sub>is the pooled standard deviation\u00a0\u00a0<\/li><li><span style=\"font-style: inherit; font-weight: inherit;\">n<sub>1<\/sub> and n<sub>2 <\/sub>are the sample sizes of the two groups<\/span><\/li><\/ul><p>By substituting values in the formula, we get:<\/p><p><img decoding=\"async\" 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up1xg\/9NBDfqjzrhPWOmvd0xqqT5gwwQ8l37X6ww8\/dH1VniVZsGCB6zeysTqKCTr89AXUF\/3999\/3Y+rT+9Wphrcetdmz91tX75rflStXupCLT6PPynOThCyeeOIJP1TO6NGj\/VAY7Kl3oksR4\/SjZTXpjz32mOtXs8oqVfhwx+cHi9reDmG1i7qNei327I4yJcgTJ064fuXQLndNdTNYU6Nm3w1HdMmhVA61f7Xu33\/\/vR+KoqlTp\/qh4eI3k+DuOg8W4dfmVNrQTVAVRLopw9atW924NGWv7lBpRdcZy86dO10\/FPpx0bp3dXVFH330kR97n67gyePMmTN+CA8C4dfG1GZMDzkaGBhw5wRFh1O6VC+NgnHWrFn+f\/nZLbc2bNhQ9843H3\/88bA2f3naybUKhb0eJqX2ejpN0tvb60qaSSXnEJ9c184Ivzam83aff\/652ykVKjoUE5X+0p6eVubqDpV87Db7tc6Tzp8\/3w\/duy9if3+\/uzuNQlfXQKsipx1O+Cu4p0+f7iqt9AOj8FbAEXId4i46hjXTUJfWlEKvFbk22Jp4ZG1uk9aOzdozqhuppjBFqRmPllGdtpk1U1E\/qZmLNRGqtW7xv1Gtpk9oPsKvw1ibN3XVAZTl6o4keYOvFk1vy9dubd3ibTHVVW\/fvOGnNoF4cDjs7TDxpjR2c1ZjtYtpzTCS6JyXDndVSZJ2risPTW+H58eOHXP9dqFlt3Z8Ur198z4WIK1GGCOD8OswukOMGm2Lan\/j5\/7s2R3xhrq1aNp58+a54ZMnT5YOvmo6j9Zu4s1bqs+rxts8ptXkxtsJNqqtJYoh\/DpQ\/JGb8StK8j67Q1cpqPZYlSpJ7flUKkyrWMnCSoCdQtvImhFduXLF9at9++23rq\/mMo3+MUE+hF8HUulPTVFE7dKqm5ZkaQysxtKq2VXTjrT3v\/jii+5eiHGaTs1a0mpz1dTG2gnWu63Wg2DLn\/awqHgbyhdeeMEP3WdX4qSt\/0g+pQ+1EX4d6n\/+53+GSiH2EHa7uqMeleZ0HbKm13kshWd1p5281mGrHrye1Nj6gw8+cH3N++WXX3bDrejdd99NXH5rQ6nlX7p0qRuOs1K3SsxqHxinQNU2U4P0pGkxwnzFBzpQde2jaoLT7jYSF2+OUq+rrtWMf6Y+SzXFeo\/uJmMPTqoER+LdYFpBfPlVs1tk+a12XJ3mp+nV13StvO6hGaV\/Kn8kdCCVXHQFiEohOsckOs9U76YMOuzLqrJj\/+qwWId8X331lbtBQ7xRtc7xqQG0SnytfL5LJbQvvvjClYDtEF3yLL9Kx7t27Ro6zFVJUaVllQyLPCoBjUf4dTidw9IhrNEOXC\/8gBBwzq\/D6b50Osdk4peeASEj\/AJgFR4A7iP8AqBzTVb6y3N1B9DJCL9A2L33sl7dAXQ6KjwABImSH4AgEX4AgkT4AQgS4QcgSIQfgCARfgCCRPgBCBLhByBIhB+AIBF+AIJE+AEIEuEHIEiEH4AgEX4AgkT4AQgS4QcgSIQfgCARfgCCRPgBCBLhByBIhB+AIBF+AIJE+AEIEuEHIEiEH4AgEX4AgkT4AQgS4QcgSIQfgCARfgCCRPgBCBLhByBIhB+AIBF+AIJE+AEIEuEHIEiEH4AgEX4AgkT4AQgS4QcgSIQfgCARfgCCRPgBCBLhByBIhB+AIBF+AIJE+KGtXL9+PTp48GC0bNmyaPLkydGoUaNcp+HVq1dHP\/30k38nUBvhh7bx5ZdfRlOmTImWL18e3bx5M9q5c2fU398f9fX1udf37NkTdXV1RRcvXnT\/B2oZdbfCDwMtbdu2bdG6deuihQsXRkeOHPFj71GJUMF448aNaNKkSdGPP\/7oXwGSUfJD29m0aZMfum\/8+PHuUFgGBgY4\/EVdhB\/axtq1ayMdqMyZM8ePGe7xxx\/3Q1F0584dPwQkI\/zQMa5cueL6Y8eOjaZNm+aGgTSEHzqC1QLL+vXrXR+ohfBD21PwLVq0yFV29PT0uMNjoB7CD23r1KlT0caNG4dqedXkZffu3f5VoDbCD21JDZvnzp0bbdmyxQWfzvPdunXLlQKBLGjnh7akUp\/cvn07+uKLL1wDZ1Ebv08\/\/ZQKD9RF+KEj6KqOefPm0cgZmXHYi46gkt5f\/vIXN6xGzlbzC6Qh\/NAxpk6d6oei6OrVq34ISEb4AQgS4Ye2obZ8quVNq9FV5YfRzQ+AWgg\/tJ133nnHDw2nW1yJgo\/aXtRDbS\/ahkp+R48edcPd3d3Rn\/70p2jMmDHRtWvXor\/+9a\/RuXPn3P38dLsr3eUFqIXwQ9vQ4e7x48ejzz77zNXoKuxEDZxnzZoVrVixYui2VkA9hB+AIHHOD0CQCD8AQSL8AASJ8AMQpIaEnz0\/lYfGAGgXpcNPz1JVswPdSWPixIl+LAC0tlLhp3uqvfTSS254wYIFrg8A7aBQ+Km0Z3fS1f3TRDeT1Djr9IBpAGhVhcJv8eLF7vmp\/f39fkwUDQ4OunHW5XmIjF2wXqazO\/sCQBalDnsvXbrk+rqekmspAbSTUuF38uRJ1y97vk8XosdLjUW6tKf4A0CSUtf2qomLanr1yMB2uKBch8cAWl+JWMqscMlPbfoUfKI7agBAOylc8lON7\/PPP8+TsgC0pcIlv9OnT7u+KjvKorYXwEgrHH7nz593\/RkzZrg+ALSTwuFntxOvtnr1avcA6Tyo7QUw0ko1dZETJ06424urAkSHr8eOHYsmTJjgXwWAFlUpNRXS3d2tipJh3dixY+9euHDBvwOtrqen51d\/Qzq6VumarXDJb8uWLUPPRtUDZCo7UnT58mUeGdhGVEpXbT0QIh5gFCidl50+fXrbNFAHGq30OT+0p6+\/\/tr1n3vuOdcHQkPJL1CqnBLVtAMhCrLkp3sNJjWUTuvKNKBWTfi4ceMKzUfTHjx40B2Wah4zZ870r5Sj+aqp0gsvvODHDKeaezVZsscTqNMylNkOmtbWQ\/NTX59R5NEH9vezAE+St+F8q9LfSuurv70tq4b1vSiq6Dzj38f4d0PD9f6WZaZtGpX8QtPb2zusVqlWN2nSJD9VMfFa8f7+fj+2vr1797rac31+X1\/f3YGBAf9KeYcPH3bLk1Qzr3H6XHXaTlrm+PbSsuSlaTStrYvmadslbwsBvdeWZeHChX7sr+k1e1+9TrXerWhwcPBuV1fX0DJqu2n7aZsVXe6i89R3xt6jbav\/27T6u2p82t+yzLTNFGT46Q+sDW47d1K3YcOGofcUZSFjneZbj76cFgxlPrsWrX9SqOuz7cuoZY9TGNt65AliCyt9uauns\/VMWpY0tuOqqxV+mqc+08I2qbOA1HAj2HI1im0ffRfj4j8AWr88is7TfgCTtrm+NxZuSX\/LMtM2U5Dhpz9CvV9NvUd\/FP1xirA\/qn3Z1GXZyWznblbwib5kSetvAZf2JbQvab1tF2frnzRN3p3YdiKbZ9LOZOrN0\/4+teaRl61LI8S3TdKPjW2DPIFRZp627dO+w\/r7ps27zLTNFGT41aM\/kv4QeXbyavoiWXjaH7Ze+Flps5E7ZDXbAapLdmLBW10qMLZzaL2yiK970ueJBarmXYstt3Yk25nKbCebR96SUy2an7pGsO+C\/iZJ4iVxbZssmjFPY9tzpKctg6YuCTZv3uz6b731luvnpdt9HTp0KNq9e3fm2\/urQkANx2XTpk2u3wzWxEXPYYnTCelz58654ccee8z1q9lNLPTQqizXb\/\/73\/\/2Q1H0m9\/8xg8NZ\/eCVIPrWl555RXXIDvPs2HSaF23bt3q5qcT8K3ItoeWMcnUqVP90P2\/aT3NmKe5cuWK61d+zHJf6FBm2jIIvyqqdVJNaKVUUeg5xNqx9DhPTZ9nx9q1a5frV36Vm3qThg8\/\/DCqlLL8\/+77\/vvv\/dDwnSDu2Wef9UNRdOfOHT+U7rvvvvNDUeqXev78+a5vTwFMoppJBfOBAwf8mHKOHz\/uPu\/VV1\/1Y1qP\/RCl3TUp\/h25deuWH6qtGfMUq8mV9evXu35WZaYti\/CrsmPHDtd\/4403XD+v1157ze1YKvVlpS+ASoqiEk5Rmk8tel07wLx58\/yY++xhVFmdOXPGD6X75ptv\/FA2SU1pVMJct26du3yyUT8KVrJ++eWXXb\/V5L0rkm4uUk8z5in6TqlZkb7z+hvlKZmXmbYRCL8Ylfr0\/GEdFlQfFmZhh7u9vb25So0qiZjZs2e7EKhuZ6cvieafRMut90+ZMiX1PWKfk3RVR55f+qxu3rzph4rTj4EOh9577z0\/phyVMgYGBtzO1qpPHMxSqs6r0fPUd3Tjxo3uO6fw6uvry\/yDX2baRiL8YqzUt2bNGtfPQ79iOtwtcl7q6tWrfihypZwVK1a4w0Qd5h0+fNgdpupQXI8NsEMEsdDTZ2q8vkg7d+70r\/7aV1995d5b5HD+QbDD3TznTuuxUt\/KlStdH\/npx3ju3LnuHLW+c\/px0o9nvSMPKTNtoxF+nja+AkR\/jKVLl\/qx2dnhbpbzUvv27Rt2dUL8EEPBpGeiaOfUYZ5KoFouO0+3fPnyodbwP\/\/8s1telWTsfIlCMunwUeLzaXV2uJv13KmuGElbb6PXta00zyIn1u3KkrTOJL1mnebR7vr7+12nH2aVoPUDtWrVquiZZ56pe3hdZtpGI\/y8\/fv3u\/DSjpa3lGGHu9qpRDtZvDM6r6b3vv3229HTTz\/txw6XVvzXNMYOXxWO77\/\/vgtMnb9SEMrHH3\/s+nH6Ymn9fve73\/kx5Tz00EN+qHFGjx7th+6f+1QpuHp7Wu2gfrD0f5V+tW5PPPGEG5\/GavE1z07y8MMP+6HGqTVPfe\/sh1nf1wsXLgz9CL\/44ov+XcnKTNtwvslL0KzBqzZH5Y\/gx2ZXCb2hdkpZO7VtMvHpa7H3pLXDi7eXql4PvaZ1TKM2iDathpNkeU9cfHnSJL0n\/jl5ulpsnpXDfj+m8bIsR1Y2r\/j3pFqW98Q1Y54m3kYwb9vJMtOWQcmv4pNPPnElBx0SFjkfVu8ZJKayAw6Ni58XzPuZ+pVMEj9ct\/OX5p\/\/\/Ge0YMEC\/79fi5e60mpyr1275oeiuqUsiZcOVUJLYqU4NfExKhnEt191V9kx3ftU0o6Pr8WaEhU5n\/sgWCk+rdbVTn3Ik08+6Ydqa8Y8Tbx5VPwcdhZlpi2D8KvYvn2767\/++uuuP9Lynn9Ka6elENV5FNH5PTuJrL7OrSxZssT9P4mWwXYOC6Rq3377resrqLKcGlDNtYkHZ9zZs2ddv1Ywl6WdWqcltH5Fzuc+CLY90p6J\/cMPP\/ihyJ0vy6IZ82xnwYefQkIlKdWCqsTxIMSbnqQ1VYmPjzc2rmZXpagkq\/OYYucI7WqKNFaxoG2SJG9bRAWqtqt89tlnrh+nUFIoy3\/\/93+7fjNYKbjI+dwHxX6o9N1Mqgj44osvXF9HK1nXqcw87RZhabWyt2\/f9kP3SuRxZaZtqsrhQtB0DkibIeu5Br1Xnc7TZaFzbzZNrfNklRJbzflqfNbPtXnZOT79P+16zrj4suo8TJyuzdV4bS+dI43T+SGbrnodtV3tterrNu1a08rO5sdkU29bxcXXScPNZJ\/TKPbdrN4+Wg\/9bfVa9TaNny9NOm9XZJ5i3z9t+yS1vp9lpm2moMPPdmj90at36DR6f54\/VPxkbq2TyPp8BZTNW6GhL7L69uXQ61mWM74DaHqtX9YT2PGw0jSal\/qah7qkHUOv2zTV4SfxMLb1snFZ18novbYDZ\/m7WcCOxI6lz1HXKNrWWkfNU2Fl34daP9jxv33S37zIPMW+gzad9h2b1r63aX\/LMtM2U9Dhpz+ENnxa7WkS+yNm2ZlsB493ab9+RmFpXwjr9FlpX8o08S+cuqTQSqMvpm0bddpZtNxpJad64SfxL7o67WyaLs8XPr7jWqf\/11o3e792uGazZWokbXNt+\/h6W2gl0Xh7X1L4Sd55iv5O+hvqffG\/o+ZR7\/tZZtpm4hkeHUq1q2pJL5UvWfTLL7+4YTSPzmsJu1R7oLa3Q6nyplK6csN57i6D4hR6BF\/7IPw6mF3R8Nvf\/tb1AdzHYS+AIFHyAxAkwg9AkAg\/AEEi\/AAEifADECTCD0CQCD8AQSL8AASJ8AMQJMIPQJAIPwBBIvwABInwAxAkwg9AkAg\/AEEi\/AAEifADECTCD0CQCD8AQSL8AASJ8AMQJMIPQJAIPwBBIvwABInwAxAkwg9AkAg\/AEEi\/AAEifADECTCD0CQCD8AQSL8AASJ8AMQJMIPQJAIPwBBIvwABInwAxAkwg9AkAg\/AEEi\/AAEifADECTCD0CQCD8AQSL8AASJ8AMQJMIPQJAIPwBBIvwABInwAxAkwg9AkAg\/AEEi\/AAEifADECTCD0CQCD8AQSL8AASJ8AMQJMIPQJAIPwBBIvwABInwAxAkwg9AkAg\/AEEi\/AAEifADECTCD0CQCD8AQSL8AASJ8INz\/fr16ODBg9GyZcuiyZMnR6NGjXKdhlevXh399NNP\/p1AZyD8EH355ZfRlClTouXLl0c3b96Mdu7cGfX390d9fX3u9T179kRdXV3RxYsX3f+BTjDqboUfRqC2bdsWrVu3Llq4cGF05MgRP\/YelQgVjDdu3IgmTZoU\/fjjj\/4VoL1R8sOQTZs2+aH7xo8f7w6FZWBggMNfdAzCD9HatWsjHQDMmTPHjxnu8ccf90NRdOfOHT8EtDfCD3VduXLF9ceOHRtNmzbNDQPtjvBDTVYLLOvXr3d9oBMQfkil4Fu0aJGr7Ojp6XGHx0CnIPzwK6dOnYo2btw4VMurJi+7d+\/2rwKdgfDDMGrYPHfu3GjLli0u+HSe79atW64UCHQS2vlhGJX65Pbt29EXX3zhGjiL2vh9+umnVHigYxB+qElXdcybN49Gzug4HPaiJpX0\/vKXv7hhNXK2ml+g3RF+qGvq1Kl+KIquXr3qh4D2RvgBCBLhB9eWT7W8aTW6qvwwuvkB0AkIPwx55513\/NBwusWVKPio7UWnoLYXruR39OhRN9zd3R396U9\/isaMGRNdu3Yt+utf\/xqdO3fO3c9Pt7vSXV6ATkD4wR3uHj9+PPrss89cja7CTtTAedasWdGKFSuGbmsFdArCD0CQOOcHIEiEH4AgEX4AgkT4AQhSQ8LPnvPKw20AtIvS4adnvqp5hO74MXHiRD8WAFpbqfDTvd9eeuklN7xgwQLXB4B2UCj8VNqzO\/7qPm+im15qnHV6EDYAtKpC4bd48WL3nNf+\/n4\/JooGBwfdOOvyPOzGLqwv09kdiAEgi1KHvZcuXXJ9XffJNZ8A2kmp8Dt58qTrlz3fpwvm46XGIt2cOXP83ACgvlLhZxfAP\/XUU67f6pIOl+noGtGh\/RQOP7XpUxMX0Z0\/AKCdFL6ri2p8n3\/+eZ7oBaAtFS75nT592vVV2VEWtb0ARlrh8Dt\/\/rzrz5gxw\/UBoJ0UDj+77Xm11atXuwdd50FtL4CRVqq2V06cOOFug64KEB2+Hjt2LJowYYJ\/FQBaU+EKDz3T4dChQ\/5\/9+iZD2r7xxO+2oNK6bosEY1RcFfCA1K45Ldly5ahZ7gq9Hp6eqLLly8TfG1EpXTV1gMh4gFGgdJ52enTp0d9fX08mQ1BKn3OD+3p66+\/dv3nnnvO9YHQUPILlCqnRDXtQIiCLPnpXoNJDaXTujINqFUTPm7cuELz0bQHDx50h6Wax8yZM\/0r5Wi+aqr0wgsv+DHDqeZelSH2eAJ1WoYy20HT2npofurrM2o9+sA+O0tnYR6nee\/bt8+9Zp+rTttR3wFthzTxbR\/fDhqut9xoEyr5haa3t1el3UzdpEmT\/FTFdHd3D82rv7\/fj61v7969d8eOHes+v6+v7+7AwIB\/pbzDhw+75blw4YIfc5\/G6XPVaTtpmePbS8uSl6bRtLYumqdtF31O0nKIfWaWrnq54svc09PjPtPWRZ+p8V1dXXcHBwf9FPdp+9h7Fi5c6P6vafUZWgeNr7XcaA9Bhp92Bn2BbedO6jZs2DD0nqIsZKzTfOvRzmjBUOaza9H6J4W6Ptt2bi17nMLY1iNPECsgNI3Cono6W8+kZdF79ZoCysInqdN81VWHmEJL0ydtQ02n19RpW1TTNHpN86imz7FgTFputI8gw09f6qQvfZzek7RTZWU7Sd6Sn3Z2vbdZwSfaaZPW3wIubae2nb7etouz9U+axoJRXXXJzQKqVtDae5K2Vb2\/n4V80rpa+KX9vbQuer3e8qG1BRl+9dhOlWcnr6ad3nY+21HqhZ+VNpNKHI1igVNdshMLXi1HEgsyrVcW8XVP+jyxQNW889J20rRFAsimzboucRaO6jj0bV80dUmwefNm13\/rrbdcPy\/d7ktXv+zevTvz7f1VIaCG47Jp0ybXbwZr4qLnsMTpBL\/dnPaxxx5z\/Wp2Ews9tCrL9dv\/\/ve\/\/VAU\/eY3v\/FDw9m9INXgOg9tL1XaVH6gCj0y1W7DVuQu5FeuXHH9SnDSqL+NEX5VVIunnapSMii0UylE9DhPTZ+n8fCuXbtcX7cIa+ZNGj788MOoUsry\/7vv+++\/90NRNHXqVD803LPPPuuHoujOnTt+KN13333nh6LUkJg\/f77r21MAs\/r4449d\/49\/\/KPr56HgrJQW3fDrr7\/u+llZLbCsX7\/e9dGeCL8qO3bscP033njD9fN67bXX3I6sUl9W2qHsOulXXnnF9YvQfGqx0t28efP8mPvsYVRZnTlzxg+l++abb\/xQNlmb0ugHStckF\/mhUIn1D3\/4gxvu6+vLNb22n5rN6O+rEmeeJxSi9RB+MbZTTZo06VeHhVnY4W5vb2+uUuPx48f9UBTNnj3bhUB1OzvtdJp\/Ei233j9lypTU94h9TtJVHbdu3fJDjXPz5k0\/1Fj2A\/Xmm2+6fj0KLW0XbSNd0qdD7f7+\/swlc\/09Nm7c6Lavgk+hmefHDa2J8IuxnWrNmjWun4d2MB3uKjjzlgiuXr3qh6Jo3bp10YoVK9xh4oEDB6LDhw+7w1QdiuuxAXbIJRZ6+kyN1465c+dO\/+qvffXVV+69RQ7nW4X9QOl8W5bwUnA98sgjbtvZHWwefvjh6Nq1a264Hv3w6OH8Oh+r7avP1Q9FvVI2Wh\/h5+nLrADRl3vp0qV+bHZ2uKvAqseuOjC6J6JRMOlk\/MqVK90hmUqgWi47T7d8+XIXAPLzzz+75dX5Kzv\/pJBMO3yMz6ddffLJJ66f9XzbE0884Up56lRi07lYlc61HfU3qBdiNq1+hHSoq9MGq1atip555pncN+1Fi\/G1vsGz5gtFmreoGYemVfOJyo7yq06vqVM7Or23EljDmpNYswt1aeJt4jSfatauUK8nrYNNr+VJEm++kfae6nWpJ8t65Wk2El\/HMu3r1KbQPjPv31vLaMuQ1EYQ7YPwqyi7U8V38qyddnqTJSTE3pPWDi8eJNXrode0jmniwZYl\/NLeExdfnjRZ3mPsvUV+oKqpXaF9bt6\/efxql+rG2WgfHPZW6FBKh6w6JCxyPqzeM0hMJTCGxsXPC+b9TB3mJokfrtv5S\/PPf\/6zZpu20aNH+6H0mtz4eTIdTtbz0EMP+aH0mlxrM6ea23q2bt3q+jolUFb8wVs6fZBHvClQ\/Hwt2gvhV7F9+3bXz9vmq1HyNpRNe2KeQlTnpUTn9+x8lvo6V7VkyRL3\/yRahkrJ0A1bIFX79ttvXV9BlaXxtmquTVoFw9mzZ12\/XmNjq9CplJJpWIzGqJRCgmbnf5p5\/kbzV5d2qKjDLntP2mVgdl6x1nwkPi8dJoqtY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alt=\"\" \/><\/p><p>\u00a0<\/p><p style=\"font-style: inherit; font-weight: inherit;\">The degree of freedom for this problem is n1 + n2 &#8211; 2, which is 25 + 30 &#8211; 2 = 53.<\/p><p>To conclude, we need to compare the calculated t-statistic (1.48) with the critical t-value at a specified significance level (0.05) and degrees of freedom (df = 53).\u00a0<\/p><p>Remember, to follow these steps:<\/p><p style=\"font-style: inherit; font-weight: inherit;\"><strong>Step 1. Find the Critical t-value:\u00a0<\/strong>Using a t-distribution table or statistical software, find\u00a0the critical t-value for a two-tailed test with a significance level of 0.05\u00a0and degrees of freedom (df) equal to 53.<\/p><p style=\"font-style: inherit; font-weight: inherit;\"><strong style=\"font-style: inherit;\">Step 2. Compare the t-statistic and Critical t-value: <\/strong><span style=\"font-style: inherit; font-weight: inherit;\">If the calculated t-statistic (1.48) is\u00a0greater than the critical t-value, we would reject the null hypothesis. If it&#8217;s\u00a0less than the critical t-value, we would fail to reject the null hypothesis.<\/span><\/p><p style=\"font-style: inherit; font-weight: inherit;\"><strong style=\"font-style: inherit;\">Step 3. Draw a Conclusion: <\/strong>Based on the following assumptions,<\/p><p style=\"font-style: inherit; font-weight: inherit;\"><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">If calculated |t| &gt;\u00a0t critical, reject the null hypothesis. This would suggest that there is a\u00a0significant difference between the average scores of the two tutoring programs.<\/span><\/p><p style=\"font-style: inherit; font-weight: inherit;\">If calculated |t| &lt;\u00a0t critical, fail to reject the null hypothesis. This would suggest that there is not enough evidence to conclude a significant difference between the average scores of the two tutoring programs.<\/p><p><span style=\"font-style: inherit; font-weight: inherit;\">(Also note that if the p-value is less than the significance level (usually 0.05), the null hypothesis is rejected. This means that there is a statistically significant difference in the average scores between Program A and Program B).<\/span><\/p><p>\u00a0<\/p><p><strong><span style=\"font-style: inherit;\">Conclusion<\/span><\/strong><\/p><p>Using a t-distribution table, the critical t-value for a two-tailed test with a significance level of 0.05 and degrees of freedom (df) equal to 53 was found to be 2.0.\u00a0<span style=\"font-style: inherit; font-weight: inherit;\">The calculated t-statistic (<\/span>1.48) is lesser than the critical t-value (2.0), we fail to reject the null hypothesis,\u00a0suggesting that there is not enough evidence to conclude a significant difference between the average scores of the two tutoring programs.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f681fcc e-flex e-con-boxed e-con e-parent\" data-id=\"f681fcc\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-abb4438 elementor-widget elementor-widget-heading\" data-id=\"abb4438\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Two-Sample t-Test Assuming Unequal Variances<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-0689592 e-flex e-con-boxed e-con e-parent\" data-id=\"0689592\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-dc5b42b elementor-widget elementor-widget-text-editor\" data-id=\"dc5b42b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The two-sample t-test assuming unequal variances is used when the variances of the two groups are not assumed to be equal. This assumption is often made when the sample sizes of the two groups are different or when the data is not normally distributed. For example, if we are comparing the heights of men and women, we can use the two-sample t-test assuming unequal variances to determine if there is a significant difference in the mean heights.<\/p><p style=\"font-style: inherit; font-weight: inherit;\">To perform this t-test, we calculate the differences between the means of the groups, and and variances of the groups. The t-value is then calculated by dividing the difference of means by the square root of the sum of the ratios of variance of the first group and the number of observations in that group and <span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">ratios of variance of the second group and the number of observations in that group<\/span><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">. This t-value is then compared to the critical value to determine if the difference is statistically significant. The two-sample test assuming unequal variances is also known as Welch&#8217;s t-test.<\/span><\/p><p style=\"font-style: inherit; font-weight: inherit;\">To perform a two-sample t-test assuming unequal variances, the researchers first need to calculate the t-score by using the formula:\u00a0<\/p><p style=\"font-style: inherit; font-weight: inherit;\"><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"17:1-17:94\"><span style=\"font-style: inherit; font-weight: inherit;\">where:<\/span><\/p><ul><li>x\u0304<sub>1<\/sub> is the mean of the first group<\/li><li>x\u0304<sub>2<\/sub> is the mean of the second group<\/li><li>\u03c3<sub>1<\/sub><sup>2 <\/sup>is the variance of the first group<\/li><li>\u03c3<sub>2<\/sub><sup>2 <\/sup>is the variance of the second group<\/li><li>n<sub>1<\/sub> is the sample size of the first group<\/li><li>n<sub>2<\/sub> is the sample size of the second group<\/li><\/ul><p data-sourcepos=\"37:1-37:11\">The variance for both groups is calculated using the following formulae:<\/p><p data-sourcepos=\"37:1-37:11\"><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/>\u00a0and\u00a0<img decoding=\"async\" style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKYAAABICAYAAAB8xo6FAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAfRSURBVHhe7Z29b9U8FIfd9x9ALVsnRLsgBgYKlRBdKewgQF3YqjIyQAs7UmGnZUBi4kPAiIQYQCqIBYRADAx8iIERivgL+uYxOcgNdm7ujZv43pxHstL4Oolj\/3zsYyfp2FaGUZTE+C\/fKkpSqDCVJFFhKkmiwlSSRIWpJIkKU0kSFaaSJCpMJUlUmEqSqDCVJFFhKkmiwlSSRIWpDMzLly\/N+fPnzfT0tBkbG7PhzJkz5v3793mKGvB0kVKdL1++bC0tLW2dPn06jxlN1tfX7T2+ePEij\/mX+fn5rZmZGVsmQFokFaNsOiHMqakpW2BVQ6gy7t69a89FpXUBygHh0RCrQvkh2Lp0QpjSkstER\/z4+HgwjZwjdPwwsbq6+rc8iqEoqh8\/ftjGePny5TwmjJTR48eP85jB6UxXTsFSaIiPwvaBJSRNUXxSOf1YjlGiSqN89+6dLVt6lRh0aoxJt0QBh8ZACNBXARQ28TKW6iKUXaiLprxouLFECZ0SJq0agRH6KUQqhRACa4y1ILii5noyvuXvtiEPocYpPUqo0Ur3X+xtKEfuO\/b9dUqYIAVMYVa1gKQPdeN0\/\/xGhXFOsSpUFPscS0gBGgn5kjKQRkT+JZ\/85oO0\/O6OH+Ue3Tgph7p0TpggVqOK9ygVEqow14JIBRPHNdhH\/AgiBSSv5EnuSSwe94nVDPUkvnLAuhJXDGW9S1U6KUxauhRiSHBCL2G6iPMUy1FyhR4bxEg+2Vbphvsphxh0cuXnwIEDJitg+\/fbt2\/tNgb79++328ximBs3bti\/6\/D69WuTicfs3r07j\/mz2iKrLFVCiMOHD5vMcprnz5\/b8kiNTi9JZtbCrKys5Hv1+f79e\/6Xn58\/f5qbN2+a48eP\/xUOy3nXrl3LU2wHYX7+\/Dnfiwf5ICRNbjk7hXRLVT1z0vbqmmWsxvgqNKaU68rKEV20jNPc7pTukrh+8lgVGf8yvu7n\/Dg4pI8xeV6FzgmTikFAoWkRH1Skb0CPgBCZOzEv0y7i8fcaH4oIOdaleJ66cM+cS0RJvji\/2+DK8ho7P71oTJjcEDcmoS1PFUFy7V6CccGq+CpFrA5BrKBYFgSHFex1n5zDJ3ryiZhiIJaa4N471+Ua7CNQuQcfHFdlFiMWjQmTm3KtAoUUw3PtB7FsZV4o+URULlQcFVPMr1RsMb0Ilm3ZtTgf5\/A1kn6tehnSWIr5IZ7r+O7BRYRdtOo7SWtdOQXRy5rEhAqhcMusAgIhja+SpHJijLHEQiEUnyir5LUppFHSlTdJa8KUimkCKdyy64lYyiyDODh1BMN1sJJF6+siQ4cmLZQPru\/rKZqgNWGGLNNOIIKrGsoEwTiT8w1aWRxXHFNSDm5Z8Df5QMQ0gtieeRW4LkOJthpHK8KkcprqxqVb7CfsVGWU5cUdIpCuythvlGn8w61MJl+6dIkGkccoyr+Urvyw\/MXLRRMTE9uWudxAmqqIKLPuMI9RFD9Bi3nlyhVz9epVu2yHOPfs2WMePHhg3rx5Y9eCT506ZdOdPHnS7N271\/5dhivKKumBZbOFhYV8L8y5c+dsHpURAmEWYYzFT4wD3Ull8Sj5jXFQP3CML5QN7Lm275hiKBuH+dJraC4MivdIvDFO6hONiLY4+SvxEpqaCuqFmycNzYdB8R6JpeSkrrV08V3Uta5i6brqUSr18To\/mbDsNjQWzKyh3bqOD49nSXq2S0tL5tmzZ3ZfUfrFK0wcHgh96kOeEZycnLRbH1+\/fq3s5ITgHL6ZgGIIPc+oDC9erxwP9\/79+2Z9fd0sLi7msX9ALFm3bcW7ubmZx25H0mRj1FresnrlHcZ26AVkrZaVB9f7xisXx6hs\/Ijjk4rzowwnQbdJBEhgigihIVTZR6Q+mlxuVEaXUn8eqyjzliJI4spESTpFqUu0tXK+k7i2toYq8xhFGZwob0ni7CBKKHrM\/aylKzsHdYSDyBuaw0AUYTIthKX0haNHj+aplDZgZoPejFkSZlqGhSjCVNLk3r17ZnZ21j6Ak\/kGeexwoMKMABP88h1yxAAMYQ4dOmTj+I2utGn27dtnnj59ai5evGiOHDmSxw4HKsya8Hjgt2\/f\/q6G3b5924rzzp075smTJ3aRgiXeW7du2d+bhE+\/1F19awsVZk0uXLiw7TtFCPTDhw82jm8OyfeMfv36ZbdKNVSYNZEPXslzBSzTIlbh1atXdss4T6mOCjMSHz9+tFueqnK\/ziZfkxt0jMdY1Z1+6xVGBRVmJDY2Nuz2xIkTdivgfACOiAvjUOYV5X0qtkzrJP8VtoZQYUaCTwaCK0C6d8aWMzMz26wonD171orx06dPdr53eXnZLlI8fPgwT\/EH5oGLc8NlYVRQYUYAK8dLekxiuwKU7v3YsWN264KIxEEC6ep\/\/\/5tt11HhRmBkABFZLt27bLWk+XA0HwmThLPuPLWaWy4JtNXwKxBlP\/1uNNkLVepCU9cUZTFbxq5b5XyGGHozVKefy0++xqD4guCxVD2TG3bNP4lDmU7fPr6+vXr5tGjR0l+C70tVJgtgheO08QKUdE56jo6xmwJposIPPEjomTOclgeS9tpVJgtgbVkKglPXibH5+bmzMGDB\/MU3Ua7ciVJ1GIqSaLCVJJEhakkiQpTSRIVppIkKkwlSVSYSpKoMJUEMeZ\/2bF53ILuWDMAAAAASUVORK5CYII=\" alt=\"\" \/><\/p><p data-sourcepos=\"37:1-37:11\">The degrees of freedom is calculated using the following formula:<\/p><p data-sourcepos=\"37:1-37:11\"><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p data-sourcepos=\"37:1-37:11\">\u00a0<\/p><p data-sourcepos=\"37:1-37:11\">\u00a0<\/p><p data-sourcepos=\"37:1-37:11\">\u00a0<\/p><p style=\"font-style: inherit; font-weight: inherit;\" data-sourcepos=\"37:1-37:11\"><strong>Example:<\/strong><\/p><p style=\"font-style: inherit; font-weight: inherit;\" data-sourcepos=\"39:1-39:246\">A researcher wants to compare the average blood pressure of patients before and after a new treatment. The researcher knows that blood pressure can vary widely from person to person, so he assumes that the variances of the two groups are unequal.<\/p><p style=\"font-style: inherit; font-weight: inherit;\" data-sourcepos=\"39:1-39:246\"><span style=\"font-style: inherit; font-weight: inherit;\">To perform a paired two-sample t-test with unequal variances, the researcher first needs to calculate the means of both groups. Then needs to calculate the variances of both the groups, followed by the degrees of freedom. Then substitute the values in the t-test formula to calculate the t-statistic.<\/span><\/p><p style=\"font-style: inherit; font-weight: inherit;\" data-sourcepos=\"41:1-41:230\"><span style=\"font-style: inherit;\"><strong>Before Treatment<\/strong><\/span><span style=\"font-style: inherit; font-weight: inherit;\">:\u00a0120, 122, 118, 130, 125,\u00a0128, 121, 123, 126, 129<\/span><\/p><p style=\"font-style: inherit; font-weight: inherit;\"><span style=\"font-style: inherit;\"><strong>After Treatment<\/strong><\/span><span style=\"font-style: inherit; font-weight: inherit;\">:\u00a0118, 120, 115, 128, 123,\u00a0126, 120, 122, 124, 127.<\/span><\/p><p style=\"font-style: inherit; font-weight: inherit;\"><span style=\"font-style: inherit;\"><strong>\u00a0<\/strong><\/span><\/p><p style=\"font-style: inherit; font-weight: inherit;\"><span style=\"font-style: inherit;\"><strong>Step 1<\/strong><\/span><span style=\"font-style: inherit; font-weight: inherit;\">: Calculate the means of both groups.<\/span><\/p><p>x\u0304<sub>1<\/sub> (mean of the first group) = 1232\/10<\/p><p>= 123.2<\/p><p>x\u0304<sub>2<\/sub> (mean of the second group) = 1213\/10<\/p><p>=121.3<\/p><p>\u00a0<\/p><p><span style=\"font-style: inherit;\"><strong>\u00a0<\/strong><\/span><\/p><p><span style=\"font-style: inherit;\"><strong>Step 2<\/strong><\/span><span style=\"font-style: inherit; font-weight: inherit;\">: Calculate the variances for both groups.<\/span><\/p><p>\u03c3<sub>1<\/sub><sup>2 <\/sup>(variance of the first group) = 160.2\/9<\/p><p>=17.8<\/p><p>\u03c3<sub>2<\/sub><sup>2 <\/sup>(variance of the second group) = 154.8\/9<\/p><p>= 17.2<\/p><p>\u00a0<\/p><p style=\"font-style: inherit; font-weight: inherit;\"><span style=\"font-style: inherit;\"><strong>\u00a0<\/strong><\/span><\/p><p style=\"font-style: inherit; font-weight: inherit;\"><span style=\"font-style: inherit;\"><strong>Step 3<\/strong><\/span><span style=\"font-style: inherit; font-weight: inherit;\">: Calculate degrees of freedom.<\/span><\/p><p>df = 17.9<\/p><p>\u00a0<\/p><p style=\"font-style: inherit; font-weight: inherit;\"><strong>\u00a0<\/strong><\/p><p style=\"font-style: inherit; font-weight: inherit;\"><strong>Step 4<\/strong>: Calculate the t-statistic by substituting the above values in the formula:<\/p><p style=\"font-style: inherit; font-weight: inherit;\"><img decoding=\"async\" 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alt=\"\" \/><\/p><p style=\"font-style: inherit; font-weight: inherit;\" data-sourcepos=\"41:1-41:230\"><img decoding=\"async\" style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\" 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4fyXZ0U+AGbwz6+s\/wug3Sxfc2J3Y7nWLanjoRZgU0ES+hSmOZMXRbIPGVfQOBMP3o7Lu3LPU7D1xJt8QuuUCAtQlihRVqyH7EVaNOApkiogPalVWFHgsj27NnjloXoG7UxK6AdYm1MIYZEwqxh9+7dzsonrx8xJKrKCpEgemMKkSASphAJImEKkSASphAJImEKkSASphAJImEKkSASphAJImEKkSASphAJImEKkSASphAJImEKkSASphDJkWX\/AuZBvlxWk4GuAAAAAElFTkSuQmCC\" alt=\"\" \/><\/p><p style=\"font-style: inherit; font-weight: inherit;\" data-sourcepos=\"41:1-41:230\"><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p style=\"font-style: inherit; font-weight: inherit;\" data-sourcepos=\"41:1-41:230\"><strong>t =\u00a0 1.02<\/strong><\/p><p data-sourcepos=\"41:1-41:230\">\u00a0<\/p><p data-sourcepos=\"41:1-41:230\">Using a t-distribution table, the critical t-value for a two-tailed test with a significance level of 0.05 and degrees of freedom (df) equal to 9 is found. If the calculated t-statistic is greater than the critical t-value, we reject the null hypothesis, suggesting that there is a significant difference between the averages compared.<\/p><p style=\"font-style: inherit; font-weight: inherit;\" data-sourcepos=\"61:1-61:241\">For the present example, the t-statistic (1.02) is lesser than the t-critical (2.262) indicating that the average of Group 1&#8217;s and Group 2&#8217;s blood pressure are not different.\u00a0<\/p><p>The researcher can conclude that the new treatment is not effective in reducing blood pressure.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f060d82 e-flex e-con-boxed e-con e-parent\" data-id=\"f060d82\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0381fb7 elementor-widget elementor-widget-heading\" data-id=\"0381fb7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">C. Paired Samples t-Test<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-aa90b9c e-flex e-con-boxed e-con e-parent\" data-id=\"aa90b9c\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-e5dc351 elementor-widget elementor-widget-text-editor\" data-id=\"e5dc351\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The\u00a0<strong>paired samples t-test is used to compare the means of two related groups of data, where the groups are paired together<\/strong>. For example, a teacher might use the paired samples t-test to compare the average pre-test and post-test scores of her students, or a doctor might use the paired samples t-test to compare the blood pressure of patients before and after a new treatment.<\/p><p>The paired two-sample t-test is used when the data is paired or matched. This means that each observation in one group is related to a specific observation in the other group. For example, if we are comparing the effectiveness of a new drug, we might measure the blood pressure of each patient before and after taking the drug. The paired t-test allows us to determine if there is a significant difference in blood pressure before and after taking the drug.<\/p><p>To perform a paired t-test, we calculate the difference between the paired observations, and then calculate the mean and standard deviation of these differences. The t-value is then calculated by dividing the mean difference by ratio of the standard deviation of the differences and square root of number of pairs. This t-value is then compared to the critical value to determine if the difference is statistically significant.<\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\"><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">To perform the paired t-test, the researchers first need to calculate the t-score by using the formula:\u00a0<\/span><\/p><p style=\"font-size: 24px;\"><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\" data-sourcepos=\"17:1-17:94\"><span style=\"font-style: inherit; font-weight: inherit;\">where:<\/span><\/p><ul style=\"font-style: normal; font-weight: 400;\"><li style=\"font-size: 24px;\">md is the mean difference between the groups<\/li><li style=\"font-size: 24px;\">SDd is the standard deviation of the differences<\/li><li><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">n<span style=\"font-size: 18px;\">\u00a0<\/span><\/span><span style=\"font-size: 24px; font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">is the number of pairs<\/span><\/li><\/ul><div>The mean difference between the groups is calculated by the following formula:<br \/><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/div><div><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\" data-sourcepos=\"17:1-17:94\"><span style=\"font-style: inherit; font-weight: inherit;\">where:<\/span><\/p><ul style=\"font-size: 24px; font-style: normal; font-weight: 400;\"><li style=\"font-size: 24px;\">md is the mean difference between the groups<\/li><li style=\"font-size: 24px;\"><span style=\"font-size: 18.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-latin; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;\">\u03a3<\/span>d is sum of the differences<\/li><li style=\"font-size: 24px;\"><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">n<\/span><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">is the number of pairs<\/span><\/li><\/ul><div>The standard deviation of differences is calculated by the following formula:<\/div><div><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/div><div><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\" data-sourcepos=\"17:1-17:94\"><span style=\"font-style: inherit; font-weight: inherit;\">where:<\/span><\/p><ul style=\"font-size: 24px; font-style: normal; font-weight: 400;\"><li style=\"font-size: 24px;\">di is the individual difference<\/li><li style=\"font-size: 24px;\">md is the mean difference\u00a0<\/li><li style=\"font-size: 24px;\"><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); color: var(--ast-global-color-3); background-color: var(--ast-global-color-5);\">n<\/span><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); color: var(--ast-global-color-3); background-color: var(--ast-global-color-5);\">is the number of pairs<\/span><\/li><\/ul><div>\u00a0<\/div><div>\u00a0<span style=\"font-weight: bold; font-style: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">Example:<\/span><\/div><p style=\"font-size: 24px;\" data-sourcepos=\"39:1-39:246\">A researcher wants to compare the average blood pressure of patients before and after a new treatment. The researcher knows that blood pressure can vary widely from person to person, so he assumes that the variances of the two groups are unequal.<\/p><p style=\"font-size: 24px;\" data-sourcepos=\"39:1-39:246\"><span style=\"font-style: inherit; font-weight: inherit;\">To perform a paired two-sample t-test, the researcher first needs to calculate the differences between After- and\u00a0<\/span><span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">Before<\/span><span style=\"font-style: inherit; font-weight: inherit; text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">-treatment blood pressures. Then needs to calculate the mean of differences, followed by the standard deviation of differences. Then substitute the values in the paired t-test formula to calculate the t-statistic.<\/span><\/p><p style=\"font-size: 24px;\" data-sourcepos=\"41:1-41:230\"><span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\">Before Treatment<\/span><\/span><span style=\"font-style: inherit; font-weight: inherit;\">:\u00a0120, 122, 118, 130, 125,\u00a0128, 121, 123, 126, 129<\/span><\/p><p style=\"font-size: 24px;\"><span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\">After Treatment<\/span><\/span><span style=\"font-style: inherit; font-weight: inherit;\">:\u00a0118, 120, 115, 128, 123,\u00a0126, 120, 122, 124, 127.<\/span><\/p><p style=\"font-size: 24px;\"><span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\">Step 1: <\/span>Calculate differences, that is, differences is After-treatment minus Before-treatment values.<\/span><\/p><p style=\"font-size: 24px;\">Differences =<img decoding=\"async\" style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p style=\"font-size: 24px;\">Differences = (118 &#8211; 120, 120 &#8211; 122, 115 &#8211; 118, 128 &#8211; 130, 123 &#8211; 125, 126 &#8211; 128, 120 &#8211; 121, 122 &#8211; 123, 124 &#8211; 126, 127 &#8211; 129)<span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\"><br \/><\/span><\/span><\/p><p style=\"font-size: 24px;\">Differences = (-2, -2, -3, -2, -2, -2, -1, -1, -2, -2)<\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\"><span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\">Step 2:\u00a0<\/span>Calculate mean of differences.<\/span><\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\"><img decoding=\"async\" style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\">Mean of Differences =\u00a0<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-2 + -2 + -3<\/span>\u00a0+\u00a0<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">-2<\/span>\u00a0+\u00a0<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">-2<\/span>\u00a0+\u00a0<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">-2<\/span>\u00a0+\u00a0<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">-1<\/span>\u00a0+\u00a0<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">-1<\/span>\u00a0+\u00a0<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">-2<\/span>\u00a0+\u00a0<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">-2) \/ 10<\/span><span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\"><br \/><\/span><\/span><\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\">Mean of Differences =\u00a0<span style=\"color: var(--ast-global-color-3); text-align: var(--text-align); background-color: var(--ast-global-color-5);\">-19<\/span><span style=\"color: var(--ast-global-color-3); text-align: var(--text-align); background-color: var(--ast-global-color-5);\">\u00a0\/ 10<\/span><span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\"><br \/><\/span><\/p><div>Mean of Differences = -1.9<\/div><div>\u00a0<\/div><div>\u00a0<\/div><div><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\"><span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\">Step 3: <\/span>Calculate the standard deviation of differences.<\/span><\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\"><img decoding=\"async\" style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\">Squared deviations = (<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-2 &#8211; (-1.9))<sup>2<\/sup> + (-2 &#8211; (-1.9))<sup>2<\/sup> + (-3 &#8211; (-1.9))<sup>2 <\/sup><\/span>+ <span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-2 &#8211; (-1.9))<sup>2<\/sup><\/span>+ <span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-2 &#8211; (-1.9))<sup>2<\/sup><\/span>+ <span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-2 &#8211; (-1.9))<sup>2<\/sup><\/span>+ <span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-1 &#8211; (-1.9))<sup>2<\/sup><\/span> + <span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-1 &#8211; (-1.9))<sup>2<\/sup><\/span>\u00a0+ <span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-2 &#8211; (-1.9))<sup>2<\/sup><\/span>+ <span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">(-2 &#8211; (-1.9))<sup>2<\/sup><\/span><span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">)<\/span><span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\"><br \/><\/span><\/span><\/p><p>Squared deviations = (0.01<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\"> + 0.01 + 1.21<\/span>+ 0.01+ 0.01+ 0.01+ 0.81+ 0.81 + 0.01+ 0.01<span style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\">)<\/span><\/p><p>Squared deviations = 2.9<\/p><p>Now calculate the Standard Deviation of differences by substituting the values in the formula.<\/p><p>Squared Deviation of differences =<\/p><p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/p><p>SDd = 0.567<\/p><\/div><div><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\"><span style=\"font-style: inherit;\"><span style=\"font-weight: bold;\">Step 4: <\/span>Calculate the t-statistic by substituting the values in the formula.<\/span><\/p><p style=\"font-size: 24px; font-style: normal; font-weight: 400;\">\u00a0<\/p><\/div><div><img decoding=\"async\" style=\"text-align: var(--text-align); background-color: var(--ast-global-color-5); color: var(--ast-global-color-3);\" 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alt=\"\" \/><\/div><div>\u00a0<\/div><div><img decoding=\"async\" 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alt=\"\" \/><\/div><div>\u00a0<\/div><div><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\" \/><\/div><div>\u00a0<\/div><\/div><div>t = -10.596<\/div><\/div><div>\u00a0<\/div><div><p data-sourcepos=\"41:1-41:230\">Using a t-distribution table, the critical t-value for a two-tailed test with a significance level of 0.05 and degrees of freedom (df) equal to 9 is found. If the calculated t-statistic is greater than the critical t-value, we reject the null hypothesis, suggesting that there is a significant difference between the averages compared.<\/p><p data-sourcepos=\"61:1-61:241\">For the present example, the t-statistic (|-10.596|) is greater than the t-critical (2.262) indicating that the average of Before Treatement and After Treatment blood pressure are significantly different.\u00a0<\/p><p>The researcher can conclude that the new treatment is very effective in reducing blood pressure.<\/p><\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-548a534 e-flex e-con-boxed e-con e-parent\" data-id=\"548a534\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-dbcedbb elementor-widget elementor-widget-button\" data-id=\"dbcedbb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"#\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Cleck here to see the video<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-191a892 e-flex e-con-boxed e-con e-parent\" data-id=\"191a892\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-edbebc8 elementor-widget elementor-widget-button\" data-id=\"edbebc8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/myknowledgehub.org\/index.php\/research-methodolgy\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Back to the Content<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-5fb5711 e-flex e-con-boxed e-con e-parent\" data-id=\"5fb5711\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Student&#8217;s t-Test I. Basic Concepts of Student&#8217;s t-Test The Student&#8217;s t-test is a statistical test that is used to determine if there is a significant difference between the means of two groups. It is a commonly used test when the sample size is small and the population standard deviation is unknown. The t-test allows researchers &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/myknowledgehub.org\/index.php\/2023\/10\/31\/research-methodology-chapter-7-3-a\/\"> <span class=\"screen-reader-text\">Research Methodology Chapter 10.2<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","footnotes":""},"categories":[1],"tags":[],"class_list":["post-1545","post","type-post","status-publish","format-standard","hentry","category-blog"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"Student&#039;s t-Test I. 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