Question

An FM rock station claims that 85% of all its listeners are college students. A statistician decides to test this claim against the suspicion that the percentage is too high. A random sample of 2000 listeners is selected from the population of all listeners and it is determined that 1655 are college students. Perform a hypothesis test to answer the question: Do the sample results support the statistician’s claim at α = 1 %?

I. State the hypotheses (in words & symbols):

H0:

Ha:

II. Determine the model to test H0 Can one use a normal distribution for the Sampling Distribution model to perform this test? Please explain.

b) Using the appropriate notation, state the MEAN & STD ERROR of the sampling distribution:

Mean:

STD Error:

III. Determine the Decision Rule: State the decision rule:

IV. Analyze the sample data: Using the appropriate notation, state the sample result of the test, p-value and the test statistic.

sample result is:

p-value is:

test statistic is:

V. State the Conclusion:

4 Can the statistician reject the rock station's claim at α = 1%? YOU MUST EXPLAIN YOUR ANSWER.

Answer 1

The sample size is N = 2000, the number of favorable cases is X = 1655, and the sample proportion is

and the significance level is α=0.01

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p=0.85 The proportion of listeners of FM rock station has 85% students.

Ha: p<0.85 The proportion of listeners of FM rock station has less than 85% students.

This corresponds to a left-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

The significance level is α=0.01, and the critical value for a left-tailed test is zc​=−2.33.

The rejection region for this left-tailed test is R = { z : z < −2.33}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = −2.818 < zc ​= −2.33, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0024, and since p = 0.0024 < 0.01, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p is less than p0​, at the α=0.01 significance level.

Therefore, there is enough evidence to claim that the proportion of listeners of FM rock station has less than 85% students at 0.01 level of significance.

Confidence Interval

The 99% confidence interval for pp is: 0.806 &lt; p &lt; 0.8490.806<p<0.849.