In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1=μ2; α=0.01


Population statistics: σ1= 52 and σ2= 68


Sample statistics: x̅1 = 5595, n1 = 156, and x̅2= 5575, n2= 216

Verified step by step guidance

1. Identify the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). Since the claim is \(\mu_1 = \mu_2\), the hypotheses are: \(H_0: \mu_1 = \mu_2\) and \(H_a: \mu_1 \neq \mu_2\) (two-tailed test).
2. Determine the significance level \(\alpha = 0.01\) and find the corresponding critical z-values for a two-tailed test. These critical values define the rejection regions for the test statistic.
3. Calculate the standard error of the difference between the two sample means using the population standard deviations and sample sizes with the formula: \(SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\)
4. Compute the test statistic (z-score) for the difference between the sample means using the formula: \(z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{SE}\) Since the null hypothesis assumes \(\mu_1 - \mu_2 = 0\), this simplifies to: \(z = \frac{\bar{x}_1 - \bar{x}_2}{SE}\)
5. Compare the calculated z-value to the critical z-values. If the test statistic falls into the rejection region (beyond the critical values), reject the null hypothesis; otherwise, fail to reject it.

Key Concepts

Hypothesis Testing for Two Population Means

This involves comparing two population means to determine if there is a statistically significant difference. The null hypothesis (H0) typically states that the means are equal (μ1 = μ2), while the alternative suggests a difference. The test uses sample data to decide whether to reject H0 at a given significance level (α).

Level of Significance (α)

The level of significance, α, is the threshold probability for rejecting the null hypothesis when it is true (Type I error). A smaller α (e.g., 0.01) means stricter criteria for rejecting H0, reducing the chance of false positives but requiring stronger evidence from the data.

Sampling Distribution and Standard Error

The sampling distribution of the difference between sample means is approximately normal if populations are normal and samples are independent. The standard error measures the variability of this difference and is calculated using population standard deviations and sample sizes, essential for computing the test statistic.