Question

2 Confidence Intervals & Hypothesis Tests

The STAT 200 course coordinator wants to estimate the proportion of all online STAT 200 students who utilize Penn State Learning’s online tutoring services by either attending a live session or viewing recordings of sessions. In a survey of 80 students during the Fall 2018 semester, 29 had utilized their services. She used bootstrapping methods to construct a 95% confidence interval for the population proportion of [0.263, 0.475]. Use this information to answer the following questions.

1. Supposed the coordinator decides that she wanted to conduct a hypothesis test instead. She wants to know if the proportion who utilize Penn State Learning’s online tutoring services is different from 0.20. What would be the appropriate null and alternative hypotheses?

2. Based on the 95% bootstrap confidence interval, would you expect the coordinator to reject or fail to reject the null hypothesis from part a at the 0.05 alpha level? Explain your reasoning.

Hint: A hypothesis test does not need to be conducted; use the confidence interval given in the question.

Using this scenario, compare and contrast confidence intervals and hypothesis testing. List at least one similarity and at least one difference.

Answer 1

(a) The Hypothesis: Since we are asked to test if the proportion is different from 0.20, therefore

H0: p = 0.20

Ha: p 0.20

(b) Since the value of 0.20 is not contained in the confidence interval [0.263,0.475], therefore the null Hypothesis H0: p = 0.20, would be rejected.

In Hypothesis Testing we calculate a sample statistic with which we find a a p value. Based on Alpha (), the significance level, we compare this sample statistic to the critical value or we compare the p value to , based on which we either reject or fail to reject the null hypothesis. The methods and criterion are as follows:

(1) Critical value approach:

Left tailed Test: If Test stat < -Critical value, Reject H0.

Right tailed Test: If Test stat > Critical value, Reject H0.

Two Tailed Test: If Test stat < -Critical value or if If Test stat > Critical value, then Reject H0.

(2) P Value Approach: If the p value is < , Reject H0.

Using the confidence Interval approach, we find an interval which is given by p ME, where the Margin of error (ME) = Critical value * Standard error. This approach uses these values to see if p, the population parameter, lies within the interval to reject or fail to reject the null hypothesis.