Ch. 8 (8.4)

The data table contains waiting times of customers at a​ bank, where customers enter a single waiting line that feeds three teller windows. Test the claim that the standard deviation of waiting times is less than 2.1 minutes, which is the standard deviation of waiting times at the same bank when separate waiting lines are used at each teller window. Use a significance level of 0.025. Assume that the sample is a simple random sample selected from a normally distributed population. Complete parts​ (a) through​ (d) below. a. Identify the null and alternative hypotheses. Choose the correct answer below. b. Compute the test statistic. c. Find the​ P-value. d. Does the use of the single line appear to reduce the variation among waiting​ times?

H0: σ=2.1 minutes HA: σ<2.1 minutes x^2= 10.34 P-value= 0.00 There is sufficient evidence to conclude that the single line reduced the variation among waiting times because H0 is rejected by the hypothesis test.

A simple random sample of 27 filtered​ 100-mm cigarettes is obtained from a normally distributed​ population, and the tar content of each cigarette is measured. The sample has a standard deviation of 0.16 mg. Use a 0.01 significance level to test the claim that the tar content of filtered​ 100-mm cigarettes has a standard deviation different from 0.20 mg, which is the standard deviation for unfiltered​ king-size cigarettes. Complete parts​ (a) through​ (d) below. a. Identify the null and alternative hypotheses. Choose the correct answer below. b. Compute the test statistic. c. Find the​ P-value. d. State the conclusion.

a. H0​: σ=0.20 mg H1: σ≠0.20 mg b. x^2= 16.64 c. P-value= 0.161 d. Fail to reject H0. There is not sufficient evidence to conclude that the tar content of filtered​100-mm cigarettes has a standard deviation different from 0.20 mg.

The piston diameter of a certain hand pump is 0.5 inch. The manager determines that the diameters are normally​ distributed, with a mean of 0.5 inch and a standard deviation of 0.005 inch. After recalibrating the production​ machine, the manager randomly selects 28 pistons and determines that the standard deviation is 0.0044 inch. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the α=0.05 level of​ significance? What are the correct hypotheses for this​ test? Calculate the value of the test statistic. Use technology to determine the​ P-value for the test statistic. What is the correct conclusion at the α=0.05 level of​ significance?

H0​: σ=0.005 H1​: σ<0.005 x^2= 20.909 P-value= .210 Since the​ P-value is greater than the level of significance, do not reject the null hypothesis. There is not sufficient evidence to conclude that the standard deviation has decreased at the 0.05 level of significance

Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than 4​%. A​ mutual-fund rating agency randomly selects 26 months and determines the rate of return for a certain fund. The standard deviation of the rate of return is computed to be 2.83​%. Is there sufficient evidence to conclude that the fund has moderate risk at the α=0.01 level of​ significance? A normal probability plot indicates that the monthly rates of return are normally distributed. What are the correct hypotheses for this​ test? Calculate the value of the test statistic. Use technology to determine the​ P-value for the test statistic. What is the correct conclusion at the α=0.01 level of​ significance?

H0​: σ=0.04 H1​: σ<0.04 x^2= 12.514 P-value= 0.018 Since the​ P-value is greater than the level of significance, do not reject the null hypothesis. There is not sufficient evidence to conclude that the fund has moderate risk at the 0.01 level of significance.

Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than 5​%. A​ mutual-fund rating agency randomly selects 22 months and determines the rate of return for a certain fund. The standard deviation of the rate of return is computed to be 4.36​%. Is there sufficient evidence to conclude that the fund has moderate risk at the α=0.05 level of​ significance? A normal probability plot indicates that the monthly rates of return are normally distributed. What are the correct hypotheses for this​ test? Calculate the value of the test statistic. Use technology to determine the​ P-value for the test statistic. What is the correct conclusion at the α=0.05 level of​ significance?

H0​: σ=0.05 H1​: σ<0.05 x^2= 15.968 P-value= 0.229 Since the​ P-value is greater than the level of significance, do not reject the null hypothesis. There is not sufficient evidence to conclude that the fund has moderate risk at the 0.05 level of significance

Data show that men between the ages of 20 and 29 in a general population have a mean height of 69.3​ inches, with a standard deviation of 2.5 inches. A baseball analyst wonders whether the standard deviation of heights of​ major-league baseball players is less than 2.5 inches. The heights​ (in inches) of 20 randomly selected players are shown in the table. Test the notion at the α=0.05 level of significance. What are the correct hypotheses for this​ test? Calculate the value of the test statistic. Use technology to determine the​ P-value for the test statistic. What is the correct conclusion at the α=0.05 level of​ significance?

H0​: σ=2.5 H1​: σ<2.5 x^2= 13.888 P-value= 0.210 Since the​ P-value is greater than the level of​ significance, do not reject the null hypothesis. There is not sufficient evidence to conclude that the standard deviation of heights of​ major-league baseball players is less than 2.5 inches at the 0.05 level of significance.

Test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the​ P-value method or the traditional method of testing hypotheses. Company A uses a new production method to manufacture aircraft altimeters. A simple random sample of new altimeters resulted in errors listed below. Use a 0.05 level of significance to test the claim that the new production method has errors with a standard deviation greater than 32.2​ ft, which was the standard deviation for the old production method. If it appears that the standard deviation is​ greater, does the new production method appear to be better or worse than the old​ method? Should the company take any​ action? What are the null and alternative​ hypotheses? Find the test statistic. Find the critical values.

H0​: σ=32.2 ft H1​: σ>32.2 ft x^2.= 35.93 Critical Values= 19.68 Since the test statistic is __greater than__ the critical​ value(s), __reject__ H0. There is __sufficient__ evidence to support the claim that the new production method has errors with a standard deviation greater than 32.2ft. The variation appears to be __greater than__ in the​ past, so the new method appears to be __worse__​because there will be __more__ altimeters that have errors.​ Therefore, the company __should__ take immediate action to reduce the variation.

The accompanying data table lists the magnitudes of 50 earthquakes measured on the Richter scale. Test the claim that the population of earthquakes has a mean magnitude greater than 1.00. Use a 0.05 significance level. Identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, P-value, and conclusion for the test. Assume this is a simple random sample. What are the null and alternative​ hypotheses? What is the test​ statistic? What is the​ P-value? Choose the correct answer below.

H0​:μ=1.00 in magnitude H1​:μ>1.00 in magnitude Z= 2.22 P-value= 0.016 Reject H0. There is sufficient evidence to conclude that the population of earthquakes has a mean magnitude greater than 1.00.

A data set lists earthquake depths. The summary statistics are n=500​, x=6.41 km, s=4.41km. Use a 0.01 significance level to test the claim of a seismologist that these earthquakes are from a population with a mean equal to 6.00. Assume that a simple random sample has been selected. Identify the null and alternative​ hypotheses, test​ statistic, P-value, and state the final conclusion that addresses the original claim. What are the null and alternative​ hypotheses? What is the test​ statistic? What is the​ P-value? State the final conclusion that addresses the original claim.

H0​:μ=6.00 km H1​:μ≠6.00 km z= 2.08 P-value= 0.038 Fail to reject H0. There is not sufficient evidence to conclude that the mean of the population of earthquake depths is 6.00 km is not correct.

Which of the following is NOT a property of the​ chi-square distribution?

The mean of the​ chi-square distribution is 0.

Thirty-seven different video games showing violence were observed. The duration times of violence (in seconds) were recorded. When using this sample for a t-test of the claim that the population mean is greater than 76 sec, what does df​ denote, and what is its​ value? What does df​ denote? What is the value of df?

The number of degrees of freedom df= 36

Which of the following is NOT a requirement for testing a claim about a standard deviation or​ variance?

The population must be skewed to the right.

Which of the following is not a requirement for testing a claim about a population with σ not​ known?

The population​ mean, μ​, is equal to 1.

Which of the following is NOT true when testing a claim about a standard deviation or​ variance?

The​ P-value method and the classical method are not equivalent to the confidence interval method in that they may yield different results.

A simple random sample of pulse rates of 50 women from a normally distributed population results in a standard deviation of 12.8 beats per minute. The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal​ range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.01 significance level to test the claim that pulse rates of women have a standard deviation equal to 10 beats per minute. Complete parts​ (a) through​ (d) below. a. Identify the null and alternative hypotheses. Choose the correct answer below. b. Compute the test statistic. c. Find the​ P-value. d. State the conclusion.

a. H0: σ=10 beats per minute H1: σ≠10 beats per minute b. x^2= 80.282 c. P-value= 0.006 d. Reject H0. There is sufficient evidence to warrant rejection of the claim that pulse rates of women have a standard deviation equal to 10 beats per minute.

The accompanying data are​ drive-through service times​ (seconds) recorded at a fast-food restaurant during dinner times. Assuming that dinner service times at the​ restaurant's competitor have standard deviation σ=64.5 ​sec, use a 0.025 significance level to test the claim that service times at the restaurant have the same variation as service times at its​ competitor's restaurant. Use the accompanying data to identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, and​ P-value. Then state a conclusion about the null hypothesis. a. Identify the null and alternative hypotheses. Choose the correct answer below. b. Compute the test statistic. c. Find the​ P-value. d. State the conclusion.

a. H0: σ=64.5 minutes HA: σ≠64.5 minutes b. x^2= 63.32 c. P-value= 0.195 d. There is not sufficient evidence to conclude that there is a difference between the waiting times in the two​ restaurants, because H0 is not rejected by the hypothesis test.

A simple random sample of 33 men from a normally distributed population results in a standard deviation of 9.5 beats per minute. The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal​ range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.10 significance level to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute. Complete parts​ (a) through​ (d) below. a. Identify the null and alternative hypotheses. Choose the correct answer below. b. Compute the test statistic. c. Find the​ P-value. d. State the conclusion.

a. H0:σ=10 beats per minute H1:σ≠10 beats per minute b. x^2= 28.88 c. P-value= 0.7495 d. Do not reject H0​, because the​ P-value is greater than the level of significance. There is insufficient evidence to warrant rejection of the claim that the standard deviation of​ men's pulse rates is equal to 10 beats per minute.

Workers at a certain soda drink factory collected data on the volumes​ (in ounces) of a simple random sample of 19 cans of the soda drink. Those volumes have a mean of 12.19 oz and a standard deviation of 0.14 oz, and they appear to be from a normally distributed population. If the workers want the filling process to work so that almost all cans have volumes between 12.01 oz and 12.69 oz, the range rule of thumb can be used to estimate that the standard deviation should be less than 0.17 oz. Use the sample data to test the claim that the population of volumes has a standard deviation less than 0.17oz. Use a 0.025 significance level. Complete parts​ (a) through​ (d) below. a. Identify the null and alternative hypotheses. Choose the correct answer below. b. Compute the test statistic. c. Find the​ P-value. d. State the conclusion.

a.H0: σ=0.17oz H1: σ<0.17oz b. x^2= 12.208 c. P-value= 0.1637 d. Do not reject H0​, because the​P-value is greater than the level of significance. There is insufficient evidence to conclude that the population standard deviation of can volumes is less than 0.17 oz.

A safety administration conducted crash tests of child booster seats for cars. Listed below are results from those​ tests, with the measurements given in hic​ (standard head injury condition​ units). The safety requirement is that the hic measurement should be less than 1000 hic. Use a 0.05 significance level to test the claim that the sample is from a population with a mean less than 1000 hic. Do the results suggest that all of the child booster seats meet the specified​ requirement? What are the null and alternative​ hypotheses? What is the test​ statistic? What is the​ P-value? State the final conclusion that addresses the original claim. What do the results suggest about the child booster seats meeting the specified​ requirement?

​H0:μ=1000 hic H1​:μ<1000 hic z= -1.696 P-value= 0.0753 Fail to reject H0. There is insufficient evidence to support the claim that the sample is from a population with a mean less than 1000 hic. There is not strong evidence that the mean is less than 1000​ hic, and one of the booster seats has a measurement that is greater than 1000 hic.