Summary
Highlights
The video introduces a problem: the mean age of college students in City X was 23. A sample of 42 students now has a mean age of 23.8. The goal is to determine if the population mean age has changed at a significance level (α) of 0.05, given a population standard deviation of 2.4. The null hypothesis (H0) is that the mean age equals 23, and the alternative hypothesis (Ha) is that the mean age is not equal to 23.
For a two-tailed test with α = 0.05, the rejection region is split into both tails, meaning 0.025 in each tail. The video explains how to find the critical Z-values using a cumulative less-than Z-table, a mean-to-Z Z-table, and even the t-distribution table (when the sample size is large). For an α of 0.05, the critical Z-values are -1.96 and +1.96.
The decision rule is to reject the null hypothesis if the calculated Z-test statistic is less than -1.96 or greater than +1.96. Using the provided sample data, the test statistic is calculated as 2.16. Since 2.16 is greater than 1.96, the null hypothesis is rejected. This indicates sufficient evidence to conclude that the mean age has changed at an α of 0.05.
The video then re-evaluates the same problem with a significance level of α = 0.02. Dividing alpha into two tails gives 0.01 in each tail. The critical Z-values for this significance level are found to be -2.33 and +2.33. The test statistic remains 2.16. Since 2.16 does not fall into the rejection region (it's not less than -2.33 and not greater than +2.33), the null hypothesis is not rejected. Therefore, at α = 0.02, there is not enough evidence to infer that the mean age has changed.