A manufacturer produces hair straighteners that have a mean life of at least 1100 hours when the production process is working properly.Based on past experience, it is known that the population standard deviation is 85 hours and the distribution of the service lives of the hair straighteners is normal.The operations manager selects a random sample of 85 hair straighteners, and the mean life of hair straighteners is recorded.

a. Suppose the population mean life is 1100 hours.What is the probability that the sample mean life will be between 1095 and 1105 hours?Keep at least 4 decimal places.(Hint: consider the sampling distribution of the mean in Ch.7 where in theory we can select many random samples of size n to build the sampling distribution with the mean of the sample mean equal to the population mean. So, based on the sampling distribution, we can estimate the probability of a randomly selected sample having a sample mean service life between 1095 and 1105 hours, assuming that we have a mean of the sample mean of 1100 hours with the population standard deviation and the sample size given in the story of the question above.)

Probability is:

b. Suppose the population mean life is 1100 hours.Based on a random sample of 85 hair straighteners,

there is a 5% chance that the mean life of hair straighteners will be above..

__________ hours

(Hint: continue to consider the sampling distribution of the mean, but we look for the sample mean value such that only 5% of the samples have their mean values above it, assuming we know the mean of the sample mean of 1100 hours)

Since the operations manager does not know the population mean, you decide to help him conduct a hypothesis test for the population mean.Based on the random sample of 85 hair straighteners that he selects, you find that the sample mean life of the hair straighteners is 1114 hours.Continue to assume that based on past experience, the population standard deviation is 85 hours and the hair straightener's service life is normally distributed.Answer the following questions related to conducting this hypothesis test:

c. What are the hypotheses that you would use to conduct a hypothesis test to see if the population mean life is different from 1100 hours?

H₀ :

H₁ :

d. State the appropriate test used and reason.Determine the appropriate test statistic.Keep at least 2 decimal places. (Hint: for calculating the test statistic, consider the sample mean value from the random sample you have found above)

Appropriate test used and reason:

Test Statistic:

e. Find the critical value(s) of the test statistic at the 0.05 level of significance.Keep at least 2 decimal places.

Critical Value

f. Based on your answer in part e, is there evidence that the population mean life of hair straighteners is different from 1100 hours (use = 0.05)?Explain by providing the decision and conclusion.

g. Compute the p-value and interpret its meaning under the condition that the null hypothesis is true.Keep at least 4 decimal places.

p-Value:

h. Construct a 95% confidence interval estimate for the population mean life of hair straighteners.Fill in the table below.Keep at least 4 decimal places.Interpret the meaning of the 95% confidence interval estimates for the population mean when we do not know the population mean.(Hint: consider the interpretation of C.I. in Ch.8)

Confidence Interval (C.I.)

Interval Lower Limit:

Interval Upper Limit:

i. Based on your answer in part h, what conclusions do you reach?That is, is there evidence that the population mean life of hair straighteners is different from 1100 hours?Explain by providing your decision and reason for your decision.