Hypothesis Testing

Hypothesis Testing is a statistical method used to make decisions or inferences about a population based on sample data. It helps us determine whether there is enough evidence in the sample to support a specific claim or hypothesis about the population. 

Steps in Hypothesis Testing

1. State the Hypotheses: Define null hypothesis: H₀​ and alternate hypothesis: H_a​.
2. Choose a Significance Level (α): Example: .
3. Select a Test and Compute the Test Statistic: Choose an appropriate test (e.g., t-test, z-test, ANOVA) and calculate the test statistic using sample data.
4. Find the P-Value or Compare to Critical Value: Determine the p-value or find the critical value from statistical tables.
5. Make a Decision:  or  ​.
6. Draw a Conclusion: Interpret the result in the context of the problem. 

Types of Hypothesis Tests

1. One-Tailed Test:
   • Tests for an effect in one direction.
   • Example: .
2. Two-Tailed Test:
   • Tests for an effect in both directions.
   • Example: .

Example:

A company claims that its light bulbs last, on average, 1,000 hours. A random sample of 30 bulbs is tested, and their mean lifetime is found to be 980 hours with a standard deviation of 50 hours. At a significance level of α=0.05, test whether there is evidence to suggest that the bulbs last less than 1,000 hours.

Null Hypothesis (​): μ=1,000 hours

Alternative Hypothesis ( ​): μ<1,000 hours 

Since the sample size is small (n<30) and the population variance is unknown, we use a t-test: