Question A manufacturer produces humidifiers that have a mean life of at least 18000 hours when the production process is working properly. Based on past experience, it is known that the population standard deviation is 800 hours and the distribution of the service lives of the humidifiers is normal. The operations manager selects a random sample of 800 humidifiers, and the mean life of humidifiers is recorded. a. Suppose the population mean life is 18000 hours. What is the probability that the sample mean life will be between 18050 and 17950 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 18050 and 17950 hours, assuming that we have a mean of the sample mean of 18000 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 18000 hours. Based on a random sample of 800 humidifiers, there is a 5% chance that the mean life of humidifiers will be above.. hours (Hint: continue to consider the sampling distribution of the mean in Ch.7, but we look for the sample mean value such that only 5% of the sample mean values are above it, assuming we know the mean of the sample mean of 18000 hours) Since the operations manager would like to know the population mean, you decide to help him conduct a hypothesis test for the population mean. Based on the random sample of 800 humidifiers that he selects, you find that the sample mean life of the humidifiers is 18017 hours. Continue to assume that based on past experience, the population standard deviation is 800 hours and the vacuum'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 18000 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 obtained 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 humidifiers is different from 18000 hours (use α = 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. (Hint: consider the definition of p-value in Ch.9 which is interpreted under the condition that the null hypothesis is true) p-Value h. Construct a 95% confidence interval estimate for the population mean life of humidifiers. Fill in the table below. Keep at least 4 decimal places. Based on your result, interpret the meaning of the 95% confidence interval estimates for the population mean when we do not know the true population mean. (Hint: consider the interpretation of C.I. in Ch.8 which is applied when the true population mean is unknown) 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 humidifiers is different from 18000 hours? Explain by providing your decision and reason for your decision.

Question A manufacturer produces humidifiers that have a mean life of at least 18000 hours when the production process is working properly. Based on past experience, it is known that the population standard deviation is 800 hours and the distribution of the service lives of the humidifiers is normal. The operations manager selects a random sample of 800 humidifiers, and the mean life of humidifiers is recorded. a. Suppose the population mean life is 18000 hours. What is the probability that the sample mean life will be between 18050 and 17950 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 18050 and 17950 hours, assuming that we have a mean of the sample mean of 18000 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 18000 hours. Based on a random sample of 800 humidifiers, there is a 5% chance that the mean life of humidifiers will be above.. hours (Hint: continue to consider the sampling distribution of the mean in Ch.7, but we look for the sample mean value such that only 5% of the sample mean values are above it, assuming we know the mean of the sample mean of 18000 hours) Since the operations manager would like to know the population mean, you decide to help him conduct a hypothesis test for the population mean. Based on the random sample of 800 humidifiers that he selects, you find that the sample mean life of the humidifiers is 18017 hours. Continue to assume that based on past experience, the population standard deviation is 800 hours and the vacuum'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 18000 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 obtained 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 humidifiers is different from 18000 hours (use α = 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. (Hint: consider the definition of p-value in Ch.9 which is interpreted under the condition that the null hypothesis is true) p-Value h. Construct a 95% confidence interval estimate for the population mean life of humidifiers. Fill in the table below. Keep at least 4 decimal places. Based on your result, interpret the meaning of the 95% confidence interval estimates for the population mean when we do not know the true population mean. (Hint: consider the interpretation of C.I. in Ch.8 which is applied when the true population mean is unknown) 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 humidifiers is different from 18000 hours? Explain by providing your decision and reason for your decision.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A manufacturer produces humidifiers that have a mean life of at least 18000 hours when the production process is working properly. Based on past experience, it is known that the population standard deviation is 800 hours and the distribution of the service lives of the humidifiers is normal. The operations manager selects a random sample of 800 humidifiers, and the mean life of humidifiers is recorded.

a. Suppose the population mean life is 18000 hours. What is the probability that the sample mean life will be between 18050 and 17950 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 18050 and 17950 hours, assuming that we have a mean of the sample mean of 18000 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 18000 hours. Based on a random sample of 800 humidifiers, there is a 5% chance
that the mean life of humidifiers will be above..		hours
(Hint: continue to consider the sampling distribution of the mean in Ch.7, but we look for the sample mean value such that only 5% of the sample mean values are above it, assuming we know the mean of the sample mean of 18000 hours)

Since the operations manager would like to know the population mean, you decide to help him conduct a hypothesis test for the population mean. Based on the random sample of 800 humidifiers that he selects, you find that the sample mean life of the humidifiers is 18017 hours. Continue to assume that based on past experience, the population standard deviation is 800 hours and the vacuum'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 18000 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 obtained 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 humidifiers is different from 18000 hours (use α = 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. (Hint: consider the definition of p-value in Ch.9 which is interpreted under the condition that the null hypothesis is true)
p-Value



h. Construct a 95% confidence interval estimate for the population mean life of humidifiers. Fill in the table below. Keep at least 4 decimal places. Based on your result, interpret the meaning of the 95% confidence interval estimates for the population mean when we do not know the true population mean. (Hint: consider the interpretation of C.I. in Ch.8 which is applied when the true population mean is unknown)
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 humidifiers is different from 18000 hours? Explain by providing your decision and reason for your decision.

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