Final answer:
A z-test is used for hypothesis testing when the population standard deviation is known. When unknown, standard error is estimated and used in the test statistic formula. For the z-test example given, we reject the null hypothesis if the test statistic is more extreme than the critical value, or the p-value is less than the significance level.
Step-by-step explanation:
When conducting a hypothetical test such as the Wald Test or z-test, where z is the test statistic calculated by z = (p − π0)/√[p(1 − p)/n], the formula utilizes the estimated standard error because the population standard deviation is unknown. The z-test is used with a normal distribution when the population standard deviation is known but in practice, this is rarely the case. When we don't know the population standard deviation, we must estimate the standard error instead, which is what is done in the formula above.
For the example given where a student is testing if Jeffrey swims the 25-yard freestyle on average in less than 16.43 seconds, the null hypothesis (μ0) is 16.43 seconds. With the sample data provided, we can calculate the test statistic and compare it to the critical value or use the p-value approach. After the test calculations, if the z-score is more extreme than the critical value or if the p-value is less than the significance level (alpha), we reject the null hypothesis.