Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data.

1. Define the null and alternative hypotheses.
2. Select the significance level (α).
3. Determine the test statistic.
4. Calculate the p-value.
5. Make a decision: Reject or fail to reject the null hypothesis.

We define: \[ H_0: \text{Null Hypothesis (no effect)} \] \[ H_1: \text{Alternative Hypothesis (effect exists)} \] Typically, we test if a parameter (like mean) differs from a hypothesized value.

Examples include Z-tests, T-tests, ANOVA, etc.

In simple linear regression, we model the relationship between a dependent variable \(Y\) and an independent variable \(X\) using the equation: \[ Y = \beta_0 + \beta_1 X + \epsilon \] where:

• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope
• \(\epsilon\) is the error term

For a one-sample t-test: \[ t = \frac{\bar{X} - \mu_0}{\frac{s}{\sqrt{n}}} \] Where: - \(\bar{X}\) is the sample mean, - \(\mu_0\) is the hypothesized population mean, - \(s\) is the sample standard deviation, and - \(n\) is the sample size.

A p-value helps us determine the strength of the evidence against the null hypothesis:

A low p-value (≤ α) suggests the data provides sufficient evidence to reject Ho.

A high p-value (> α) indicates insufficient evidence to reject Ho.

# Example of conducting a t-test in R
set.seed(123)
data <- rnorm(30, mean = 5, sd = 2)
t.test(data, mu = 5)

## 
##  One Sample t-test
## 
## data:  data
## t = -0.26299, df = 29, p-value = 0.7944
## alternative hypothesis: true mean is not equal to 5
## 95 percent confidence interval:
##  4.173147 5.638438
## sample estimates:
## mean of x 
##  4.905792

• Hypothesis Testing allows us to make inferences about population parameters using sample data.

  • It involves formulating null and alternative hypotheses, selecting a significance level, calculating a test statistic, and making a decision based on the p-value.

• Simple Linear Regression helps in understanding and modeling the relationship between two quantitative variables.

  • The regression equation Y=β0​+β1​X+ϵ quantifies this relationship, allowing for predictions and insights.