• Hypothesis testing is a statistical method used to evaluate a claim about a population.
• We begin with two competing statements:
  • Null hypothesis: \(H_0\)
  • Alternative hypothesis: \(H_a\)
• Sample data is used to decide whether there is enough evidence against \(H_0\).

• The null hypothesis usually represents “no effect” or “no difference.”
• The alternative hypothesis represents the claim we want evidence for.
• We calculate a test statistic and a p-value.
• A small p-value suggests the observed result would be unlikely if \(H_0\) were true.

For a one-sample mean test:

\[ H_0: \mu = 10 \]

\[ H_a: \mu \neq 10 \]

We compute a test statistic:

\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \]

Where: - \(\bar{x}\) = sample mean - \(\mu\) = hypothesized mean - \(s\) = sample standard deviation - \(n\) = sample size

• We calculate a p-value
• The p-value tells us how likely our result is under \(H_0\)

Decision:

\[ \text{Reject } H_0 \text{ if } p < 0.05 \]

• Small p-value means stronger evidence against \(H_0\)
• Large p-value means not enough evidence to reject \(H_0\)

We test whether students study more than 10 hours per week.

t.test(study_hours, mu = 10)

## 
##  One Sample t-test
## 
## data:  study_hours
## t = 2.9274, df = 29, p-value = 0.006586
## alternative hypothesis: true mean is not equal to 10
## 95 percent confidence interval:
##  10.33180 11.87036
## sample estimates:
## mean of x 
##  11.10108

##        t 
## 2.927376

## [1] 0.006586154

## [1] 11.10108

• Hypothesis testing helps evaluate claims using data.
• We tested whether students study more than 10 hours per week.
• The graphs and test results help support the conclusion.
• This method is widely used in science, business, and education.

• It supports evidence-based decision making.
• It helps researchers and analysts test claims objectively.
• It is one of the most important tools in statistics.