Question

Suppose that you are testing the hypotheses H0​: μ =11 vs. HA​: μ <11 A sample of size 64 results in a sample mean of 11.5 and a sample standard deviation of 2.4 ​a) What is the standard error of the​ mean? ​b) What is the critical value of​ t* for a 99 % confidence​interval? ​c) Construct a 99​%confidence interval for μ. ​d) Based on the confidence​ interval, at a =0.005 can you reject H0​? Explain.

2)Before lending someone​ money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the​ applicant, totaling points they award for the​ person's income​ level, credit​ history, current debt​ burden, and so on. The higher the point​ total, the more convinced the bank is that​ it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid​ loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the​ person's score falls below the minimum cutoff will the bank reject the null and deny the loan.

​a) When a person defaults on a​ loan, which type of error did the bank​ make? A. Type I error B. Type II error

​b) Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid​ it? A. Type I error B. Type II error ​

c) Suppose the bank decides to lower the cutoff score from 250 points to 200. Is that analogous to choosing a higher or lower value of alpha for a hypothesis​ test? A. higher alpha level B. lower alpha level

​d) What impact does this change in the cutoff value have on the chance of each type of​ error? A. Increased Type​ I, decreased Type II. B. Decreased Type​ I, increased Type II. C. Increased Type​ I, increased Type II. D. Decreased Type​ I, decreased Type II

3)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 56.4​, 60.8​, 46.7​, 54.4​, 47.9​, 46.9​, 54.4​, and 40.8 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 alphaequals0.05. What are the null and alternative​ hypotheses?

b)What is the value of the test​ statistic?

c)What is the​ P-value of the test​ statistic? d)Should they market the​ wax? A. No. There is sufficient evidence to conclude the mean time is less than 55 seconds. B. Yes. There is insufficient evidence to conclude the mean time is less than 55 seconds. C. Yes. There is sufficient evidence to conclude the mean time is less than 55 seconds. D. No. There is insufficient evidence to conclude the mean time is less than 55 seconds. ​

d) Suppose they decide not to market the wax after the​ test, but it turns out that the wax really does lower the​ champion's average time to less than 55 seconds. What kind of error have they​ made? Explain the impact to the company of such an error.

A. They have made a Type II error and will lose the potential profit from selling the wax.

B. They have made a Type I error and customers might demand their money back.

C. They have made a Type I error and will lose the potential profit from selling the wax.

D. They have made a Type II error and customers might demand their money back.

Answer 1

1)

R-commands and outputs:

#H0​: μ =11 vs. HA​: μ <11
#it is one-tailed test as HA:μ <11.
mu0=11
xbar=11.5
n=64
s=2.4 #sample standard deviation

#​a) What is the standard error of the​ mean?
SEofmean=s/n
SEofmean
[1] 0.0375

#​b) What is the critical value of​ t* for a 99 % confidence ​nterval?
#For 99% Confidence interval,alpha=0.01
alpha=0.01
tstar=qt(1-alpha/2,df=n-1)
tstar
[1] 2.656145

# c) Construct a 99​%confidence interval for μ.
Low=xbar-tstar*SEofmean
Low
[1] 11.40039
Upp=xbar+tstar*SEofmean
Upp
[1] 11.59961
# 99% confidence interval is [11.40039, 11.59961]

#​d) Based on the confidence​ interval, at a =0.005 can you reject H0​?
# YES,We observe that, mu0=11 does not lie in this interval. Therefore, Reject H0.