Question

A company is developing a new​ high-performance wax for cross country ski racing. In order to justify the price marketing​ wants, the wax needs to be very fast.​ Specifically, the mean time to finish their standard test course should be less than 55 seconds for a former Olympic champion. To test​ it, the champion will ski the course 8 times. The​ champion's times are 55.7​, 60.5​, 50.8​, 54.5​, 49.2​, 47.1​, 51.1​, and 42.9 seconds to complete the test course. Complete parts a and b below.

a) Should they market the wax? Assume the assumptions and conditions for appropriate hypothesis testing are met for the sample. Assume a = 0.05. What are the null and alternative hypotheses? Choose the correct answer below. A. Ho: u = 55 vs. Ha: u <55 O B. Ho: p = 55 vs. Ha: p> 55 O C. Ho:p> 55 vs. Ha: y = 55 O D. Ho: u <55 vs. Ha: j = 55 What is the value of the test statistic? The test statistic is und to two decimal places as needed.) Calculate the P-value. P-value = (Round to four decimal places as needed.)

Answer 1

Solution:

Here, we have to use one sample t test for the population mean.

The null and alternative hypotheses are given as below:

H0: µ = 55 versus Ha: µ < 55

This is a lower tailed test.

The test statistic formula is given as below:

t = (Xbar - µ)/[S/sqrt(n)]

From given data, we have

µ = 55

Xbar = 51.475

S = 5.436582435

n = 8

df = n – 1 = 7

α = 0.05

Critical value = -1.8946

(by using t-table or excel)

t = (Xbar - µ)/[S/sqrt(n)]

t = (51.475 – 55)/[ 5.436582435/sqrt(8)]

t = -1.8339

Test statistic = -1.83

P-value = 0.0550

(by using t-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that the mean time is less than 55 seconds.

Answer:

D. No, do not market the wax as there is insufficient evidence to conclude that the mean time is less than 55 seconds.

Part b

Answer:

Here, we do not reject the null hypothesis, so we are making type II error.

Answer: C. They have made a type II error and will lose the potential profit from selling the wax.