Answer:
To perform the x-test and interpret the results, we need to compare the proportion of seniors who have taken at least one semester of a foreign language to the proportion of the entire student body who have taken at least one semester of a foreign language.
Let's start by calculating the proportion of seniors who have taken at least one semester of a foreign language. We have a sample of 80 seniors, and out of those, 52 have taken at least one semester of a foreign language. So the proportion can be calculated as:
Proportion of seniors = Number of seniors who have taken at least one semester of a foreign language / Total number of seniors
Proportion of seniors = 52 / 80
Next, let's calculate the proportion of the entire student body who have taken at least one semester of a foreign language. We know that among the entire student body of 400 students, 230 have taken at least one semester of a foreign language. So the proportion can be calculated as:
Proportion of entire student body = Number of students who have taken at least one semester of a foreign language / Total number of students
Proportion of entire student body = 230 / 400
Now, we can perform the x-test. The x-test compares the proportions and determines if they are significantly different. The formula for the x-test statistic is:
x = (Proportion of seniors - Proportion of entire student body) / √((Proportion of entire student body * (1 - Proportion of entire student body)) / Total number of seniors)
After calculating the x-test statistic, we can compare it to the critical value at the chosen significance level (alpha) to determine if we should reject or not reject the null hypothesis.
In this case, the significance level (alpha) is given as 0.05, so we need to compare the x-test statistic to the critical value of 1.96 (corresponding to alpha = 0.05).
Now, let's perform the calculations and interpret the results:
1. Calculate the x-test statistic:
x = (Proportion of seniors - Proportion of entire student body) / √((Proportion of entire student body * (1 - Proportion of entire student body)) / Total number of seniors)
2. Compare the x-test statistic to the critical value:
If the x-test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.
Based on the given options, the correct answer is D. The z-score is -1.357. You should not reject the null hypothesis.
Explanation:
<3