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=0.05


Population statistics: σ1= 14 and σ2= 15


Sample statistics: x̅1 = 87, n1 = 410, and x̅2= 85, n2= 340

Verified step by step guidance

1. Identify the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\) based on the claim. Since the claim is \(\mu_1 \neq \mu_2\), set \(H_0: \mu_1 = \mu_2\) and \(H_a: \mu_1 \neq \mu_2\).
2. Determine the significance level \(\alpha = 0.05\) and note that this is a two-tailed test because the alternative hypothesis is not equal to (\(\neq\)).
3. Calculate the standard error of the difference between the two sample means using the formula: \(SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\) where \(\sigma_1 = 14\), \(\sigma_2 = 15\), \(n_1 = 410\), and \(n_2 = 340\).
4. Compute the test statistic (z-score) using the formula: \(z = \frac{\bar{x}_1 - \bar{x}_2}{SE}\) where \(\bar{x}_1 = 87\) and \(\bar{x}_2 = 85\) are the sample means.
5. Find the critical z-values for a two-tailed test at \(\alpha = 0.05\), which correspond to the z-scores that cut off the upper and lower 2.5% of the standard normal distribution. Compare the calculated test statistic to these critical values to decide whether to reject or fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing for Two Population Means

This involves testing whether there is a statistically significant difference between two population means (μ1 and μ2). The null hypothesis typically states that the means are equal (μ1 = μ2), while the alternative reflects the claim (μ1 ≠ μ2). The test uses sample data to decide whether to reject the null hypothesis at a given significance level (α).

Level of Significance (α)

The level of significance, α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). Commonly set at 0.05, it defines the threshold for deciding if the observed difference is statistically significant. If the p-value is less than α, the null hypothesis is rejected.

Sampling Distribution and Standard Error

The sampling distribution of the difference between sample means is approximately normal when samples are large or populations are normal. The standard error measures the variability of this difference and is calculated using population standard deviations and sample sizes. It is essential for computing the test statistic in hypothesis testing.