Question

According to the IRS, taxpayers calling the IRS in 2017 waited 13 minutes on average for an IRS telephone assister to answer. Do callers who use the IRS help line early in the day have a shorter wait? Suppose a sample of 50 callers who placed their calls to the IRS in the first 30 minutes that the line is open during the day have a mean waiting time of 11 minutes before an IRS telephone assister answers. Based on data from past years, you can assume that the standard deviation of waiting times is 8 minutes. Using these sample results, can you conclude that the waiting time for calls placed during the first 30 minutes the IRS help line is open each day is significantly less that the overall mean waiting time of 13 minutes? Use alpha=0.05

Establish the two hypotheses
Calculate the test statistic
Make a decision using the critical value approach
Make a decision using the p-value approach

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 13
Alternative Hypothesis, Ha: μ < 13

Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (11 - 13)/(8/sqrt(50))
z = -1.77

Rejection Region
This is left tailed test, for α = 0.05
Critical value of z is -1.645.
Hence reject H0 if z < -1.645

reject the null hypothesis.

P-value Approach
P-value = 0.0384
As P-value < 0.05, reject the null hypothesis.