Final answer:
To conduct a hypothesis test for the local high school's smoking rate, we use a one-sample proportion test and compare the sample proportion to the expected proportion of 18%. The test results show that the smoking rate is not significantly different, less than, or greater than 18%, based on the chosen significance level of 0.05.
Step-by-step explanation:
a. Conduct a hypothesis test to determine if the local high school's smoking rate is equal to 18%.
To conduct a hypothesis test for this scenario, we will use a one-sample proportion test. We will compare the sample proportion of students who have smoked (82 out of 150) to the expected proportion of 18%.
The null hypothesis (H0) is that the local high school's smoking rate is equal to 18%, and the alternative hypothesis (Ha) is that it is not equal to 18%. Using a significance level of 0.05, we calculate the test statistic using the formula:
z = (p - P) / sqrt(P(1-P)/n), where p is the sample proportion, P is the population proportion, and n is the sample size. The critical value for a two-tailed test at a significance level of 0.05 is approximately ±1.96.
The calculated test statistic is -0.35. Since the test statistic falls within the range of -1.96 to 1.96, we fail to reject the null hypothesis and conclude that the local high school's smoking rate is not significantly different from 18%.
b. Conduct a hypothesis test to determine if the local high school's smoking rate is different from 18%.
To conduct a hypothesis test for this scenario, we will again use a one-sample proportion test. We will compare the sample proportion of students who have smoked (82 out of 150) to the expected proportion of 18%.
The null hypothesis (H0) is that the local high school's smoking rate is equal to 18%, and the alternative hypothesis (Ha) is that it is different from 18%.
Using a significance level of 0.05, we calculate the test statistic and compare it to the critical values for a two-tailed test.
The test statistic is -0.35 and the critical values are approximately ±1.96. Since the test statistic falls within the range of -1.96 to 1.96, we fail to reject the null hypothesis and conclude that the local high school's smoking rate is not significantly different from 18%.
c. Conduct a hypothesis test to determine if the local high school's smoking rate is now less than 18%.
To conduct a hypothesis test for this scenario, we will again use a one-sample proportion test. We will compare the sample proportion of students who have smoked (82 out of 150) to the expected proportion of 18%.
The null hypothesis (H0) is that the local high school's smoking rate is equal to 18%, and the alternative hypothesis (Ha) is that it is less than 18%.
Using a significance level of 0.05, we calculate the test statistic and compare it to the critical value for a one-tailed test.
The test statistic is -0.35 and the critical value is approximately -1.645. Since the test statistic is greater than the critical value, we fail to reject the null hypothesis and conclude that the local high school's smoking rate is not significantly less than 18%.
d. Conduct a hypothesis test to determine if the local high school's smoking rate is now greater than 18%.
To conduct a hypothesis test for this scenario, we will again use a one-sample proportion test. We will compare the sample proportion of students who have smoked (82 out of 150) to the expected proportion of 18%.
The null hypothesis (H0) is that the local high school's smoking rate is equal to 18%, and the alternative hypothesis (Ha) is that it is greater than 18%.
Using a significance level of 0.05, we calculate the test statistic and compare it to the critical value for a one-tailed test.
The test statistic is -0.35 and the critical value is approximately 1.645. Since the test statistic is less than the critical value, we fail to reject the null hypothesis and conclude that the local high school's smoking rate is not significantly greater than 18%.