Research Methodology Chapter 9.1

Statistical inference is a fundamental concept in research that allows us to conclude a population based on a sample. It involves using data from a sample to make inferences or predictions about the larger population from which the sample was drawn. This process is crucial in research as it enables us to make informed decisions, test hypotheses, and gain insights into various phenomena.

Statistical Models

A statistical model is a mathematical representation of the relationship between variables in a dataset. It helps researchers understand the underlying structure and patterns in the data. There are various types of statistical models, including linear regression models, logistic regression models, and ANOVA models, among others. These models provide a way to describe and analyze the data, making it easier to draw meaningful conclusions.

In a statistical model, variables can be classified into two types: dependent variables and independent variables. 

The dependent variable is the outcome or response variable that researchers are interested in studying, while independent variables are the factors that may influence the dependent variable. By including relevant independent variables in the model, researchers can assess their impact on the dependent variable and make predictions.

Estimation

Estimation is a fundamental aspect of statistical inference. It involves using sample data to estimate unknown population parameters. Population parameters are numerical characteristics of a population, such as the mean, standard deviation, or proportion. Since it is often impractical or impossible to collect data from an entire population, researchers rely on samples to estimate these parameters.

There are different methods of estimation, depending on the type of data and the population parameter of interest. For example, if the population parameter of interest is the mean, researchers can use the sample mean as an estimate. This is known as point estimation. However, point estimates are subject to sampling variability, meaning they may vary from one sample to another. To account for this variability, researchers often provide a measure of uncertainty associated with the estimate, such as a confidence interval.

Confidence intervals provide a range of plausible values for the population parameter. They are constructed based on the sample data and the desired level of confidence. For example, a 95% confidence interval for the population mean would provide a range within which we can be 95% confident that the true population mean lies. The width of the confidence interval depends on the sample size and the variability of the data.

Statistical Hypothesis

In statistical inference, hypotheses play a crucial role in the process of hypothesis testing and estimation. A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true. It serves as the basis for making inferences and drawing conclusions from sample data. 

Hypothesis testing refers to the formal procedures used by statisticians and researchers to accept or reject statistical hypotheses. There are two types of hypotheses are used in statistical analysis.

Null Hypothesis (H₀)

The null hypothesis, denoted by H₀, is usually the hypothesis that sample observations result purely from chance. It is a statement of no effect or no difference. It represents the status quo or the assumption that there is no relationship or difference between variables in the population. 

The null hypothesis is typically the hypothesis that researchers aim to reject or disprove through statistical analysis. It is often formulated as an equality statement, such as “the mean is equal to a specific value” or “there is no association between variables.”

For example, if a researcher wants to investigate whether a new drug has an effect on reducing blood pressure, the null hypothesis would state that “the mean blood pressure of individuals who receive the drug is equal to the mean blood pressure of individuals who receive a placebo.”

Alternative Hypothesis (H_a or H₁)

It is also known as the research hypothesis and is the complement of the null hypothesis. It represents the claim or assertion that researchers aim to support or establish through statistical evidence. The alternative hypothesis can take different forms depending on the research question and the nature of the study. The alternative hypothesis, denoted by H_a or H₁, is the hypothesis that sample observations are influenced by some non-random cause.

There are two common types of alternative hypotheses:

1 One-Sided (or One-Tailed) Alternative Hypothesis: In this type of alternative hypothesis, the researcher specifies the direction of the effect or difference. It states that there is an increase or decrease in the population parameter. For example, “the mean blood pressure of individuals who receive the drug is less than the mean blood pressure of individuals who receive a placebo.”

2 Two-Sided (or Two-Tailed) Alternative Hypothesis: In a two-sided alternative hypothesis, the researcher does not specify the direction of the effect or difference. It states that there is a difference between the population parameter and the hypothesized value.
For example, “the mean blood pressure of individuals who receive the drug is different from the mean blood pressure of individuals who receive a placebo.”

Example: Suppose we wanted to determine whether a coin was fair and balanced. A ull hypothesis might be that half the flips would result in Heads and half in Tails. The alternative hypothesis might be that the number of Heads and Tails would be different. 

Symbolically, these hypotheses would be expressed as:

H₀: P = 0.5

H_a: P ≠ 0.5

Suppose we flipped the coin 50 times, resulting in 40 Heads and 10 Tails. Given this result, we would be inclined to reject the null hypothesis. We would conclude, based on the evidence, that the coin was probably not fair and balanced.