Hypothesis Testing in Statistics

Null and Alternative Hypotheses

In statistical hypothesis testing, we begin by formulating two competing statements about a population parameter:

• Null Hypothesis (H0): The default assumption that there is no effect or no difference. It is the hypothesis that the test seeks to disprove.

• Alternative Hypothesis (Ha): The statement that contradicts the null hypothesis, representing the effect or difference the researcher suspects or wants to prove.

Example: Testing whether a coin is fair:

• H0: p = 0.5 (the probability of heads is 0.5)

• Ha: p ≠ 0.5 (the probability of heads is not 0.5)

Significance Level and p-Value

The significance level (denoted by ) is the threshold for deciding whether to reject the null hypothesis. The p-value is the probability, under H0, of obtaining a result at least as extreme as the observed data.

• If p-value < , reject H0.

• If p-value > , fail to reject H0.

Common significance levels: , ,

One-Sided vs. Two-Sided Tests

Hypothesis tests can be one-sided or two-sided, depending on the alternative hypothesis:

• One-sided test: Ha: parameter > value or parameter < value

• Two-sided test: Ha: parameter ≠ value

The choice affects the calculation of the p-value:

• One-sided p-value: Probability of observing a test statistic as extreme or more extreme in one direction.

• Two-sided p-value: Probability of observing a test statistic as extreme or more extreme in either direction (both tails).

Calculating p-Values for Z-Scores

The z-score measures how many standard deviations an observed proportion is from the hypothesized proportion. The p-value is determined using the standard normal distribution.

• One-sided p-value: or

• Two-sided p-value:

Example: If , the one-sided p-value is , and the two-sided p-value is .

One-Proportion Z-Test

Assumptions for One-Proportion Z-Test

Before performing a one-proportion z-test, certain assumptions must be met:

• Random sample: The data should be collected randomly.

• Independence: Observations must be independent.

• Sample size: Both and , where is sample size and is the hypothesized proportion.

Performing a One-Proportion Z-Test

The one-proportion z-test is used to test hypotheses about a population proportion.

• Step 1: State H0 and Ha.

• Step 2: Check assumptions.

• Step 3: Calculate the test statistic:

• Step 4: Find the p-value using the standard normal distribution.

• Step 5: Compare p-value to and state the conclusion.

Example: In a sample of 100 students, 60 say they prefer online learning. Test if the proportion preferring online learning is different from 0.5.

• H0: p = 0.5

• Ha: p ≠ 0.5

• , ,

• Two-sided p-value:

Stating Conclusions from Hypothesis Tests

After performing the test, interpret the results:

• If p-value < , reject H0: There is sufficient evidence to support Ha.

• If p-value > , fail to reject H0: There is not sufficient evidence to support Ha.

Example: If p-value = 0.045 and , reject H0.

Recommended Problems and Applications

Practice Problems (4th Edition)

• 16.11: Determine null and alternative hypotheses.

• 16.13: Interpret the result of a hypothesis test.

• 16.19: Find the mistakes in hypothesis testing procedures.

• 16.35: Perform and interpret a hypothesis test.

• 16.41: Understand the concept of causation in statistical inference.

Summary Table: One-Proportion Z-Test Steps

Additional info: The above notes expand on the learning objectives and recommended problems, providing definitions, formulas, and examples for self-contained study.