Two Means - Known Variance: Videos & Practice Problems

When comparing two means, the key difference between a z-test and a t-test lies in whether the population standard deviations are known. If the population standard deviations are known, we use a z-test, which relies on the normal distribution. The formula for the z-test statistic is $z=\frac{\mu }{}$₁−μ₂σ2n₁+σ2n₂. In contrast, if the population standard deviations are unknown, we estimate them using sample standard deviations and perform a t-test, which uses the t-distribution and accounts for additional uncertainty. The t-test is more appropriate for smaller samples or when population variances are unknown. Thus, knowing the population variances allows for a more precise z-test.

To perform a hypothesis test for two means with known population standard deviations, follow these steps: First, state the null hypothesis $H{0}_{}:\mu$₁=μ₂ and the alternative hypothesis depending on the claim (e.g., $H{a}_{}:\mu$₁<μ₂). Next, collect sample means, sample sizes, and known population standard deviations. Calculate the z-test statistic using the formula $z=\frac{\mu }{}$₁−μ₂σ2n₁+σ2n₂. Then, find the p-value corresponding to the z-score. Finally, compare the p-value to the significance level $\alpha$. If the p-value is less than $\alpha$, reject the null hypothesis; otherwise, fail to reject it. Ensure assumptions like independent random samples and normality or large sample sizes are met.

When conducting a two-sample z-test for means with known variances, several assumptions must be satisfied to ensure valid results. First, the samples must be independent random samples from their respective populations. This means the selection of one sample does not influence the other. Second, the populations from which the samples are drawn should be normally distributed. However, if the sample sizes are large (typically greater than 30), the Central Limit Theorem allows us to relax this assumption because the sampling distribution of the sample mean will approximate normality. Third, the population standard deviations must be known and constant. Meeting these assumptions allows the use of the z-test formula and ensures the accuracy of the hypothesis test conclusions.

The p-value in a two-sample z-test for means with known variances represents the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true. A small p-value indicates that such an extreme result is unlikely under the null hypothesis, providing evidence against it. For example, if the p-value is less than the chosen significance level $\alpha$ (commonly 0.05), we reject the null hypothesis and conclude there is sufficient evidence to support the alternative hypothesis. Conversely, a large p-value suggests insufficient evidence to reject the null hypothesis. Thus, the p-value helps determine whether the observed difference between sample means is statistically significant.

No, a two-sample z-test for means is not appropriate if the population standard deviations are unknown. The z-test requires known population standard deviations to calculate the standard error accurately. When these are unknown, we estimate the standard deviations using sample standard deviations and use a two-sample t-test instead. The t-test accounts for the additional uncertainty from estimating the standard deviations and uses the t-distribution, which has heavier tails than the normal distribution. This adjustment is especially important for smaller sample sizes. Therefore, when population variances are unknown, the two-sample t-test is the correct method for hypothesis testing on means.