Question 1

Which of the following is an example illustrating the use of variance?
A)As a measure of risk.
B)As a judge of consistency.
C)To search for and reduce variability in a process.
D)All of these choices are true.

Accepted Answer

Variance is a statistical measure used to quantify the amount of variation or dispersion of a set of data points around their mean value. It has several applications, including as a measure of risk in finance and investments, as a judge of consistency in quality control, and to search for and reduce variability in a process in manufacturing and operations management. Therefore, all choices are true and illustrate the use of variance.

Question 2

Mystic Pizza 
Mystic Pizza in Mystic,Connecticut,advertises that they deliver your pizza within 15 minutes of placing an order or it is free.A sample of 25 customers is selected at random.The average delivery time in the sample was 13 minutes with a sample standard deviation of 4 minutes. ​ ​
-{Mystic Pizza Narrative} What is the required condition of the technique used in the previous question?

Accepted Answer

The answer of Mystic Pizza 
Mystic Pizza in Mystic,Connecticut,advertises that...

Question 3

Energy Drink Consumption
 A researcher at the University of Washington medical school believes that energy drink consumption may increase heart rate.Suppose it is known that heart rate (in beats per minute)is normally distributed with an average of 70 bpm for adults.A random sample of 25 adults was selected and it was found that their average heartbeat was 73 bpm after energy drink consumption,with a standard deviation of 7 bpm. ​ ​
-{Energy Drink Consumption Narrative} Test the hypotheses in the previous question at the 10% significance level to determine if we can infer that energy drink consumption increases heart rate.

Accepted Answer

The answer of Energy Drink Consumption
 A researcher at the...

Question 4

Two uses for the population variance are to measure risk and consistency.

Accepted Answer

The population variance is commonly used as a measure of risk in finance and statistics, and it is also used to measure the consistency and variability of data points in a population.

Question 5

The area to the right of a certain critical value on a chi-squared distribution is 0.025.For 5 degrees of freedom,the critical value is 11.143.

Accepted Answer

The answer of The area to the right of a...

Question 6

A left-tailed area in the chi-squared distribution equals 0.95.For 6 degrees of freedom the critical value equals 12.592.

Accepted Answer

The critical value for a left-tailed area of 0.95 and 6 degrees of freedom in the chi-squared distribution is indeed 12.592.

Question 7

Employees in a large company are entitled to 15-minute water breaks.A random sample of the duration of water breaks for 10 employees was taken with the times shown as: 12,16,14,18,21,17,19,15,18,and 16.Assuming that the times are normally distributed,is there enough evidence at the 5% significance level to indicate that on average employees are taking longer water breaks than they are entitled to?

Accepted Answer

The answer of Employees in a large company are entitled...

Question 8

The statistic s² is an unbiased and consistent estimator of σ².

Accepted Answer

The sample variance s² is an unbiased estimator of the population variance σ², meaning its expected value equals σ². It is also consistent, as it converges to σ² as the sample size increases.

Question 9

Energy Drink Consumption
 A researcher at the University of Washington medical school believes that energy drink consumption may increase heart rate.Suppose it is known that heart rate (in beats per minute)is normally distributed with an average of 70 bpm for adults.A random sample of 25 adults was selected and it was found that their average heartbeat was 73 bpm after energy drink consumption,with a standard deviation of 7 bpm. ​ ​
-{Energy Drink Consumption Narrative} Formulate the null and alternative hypotheses.

Accepted Answer

The answer of Energy Drink Consumption
 A researcher at the...

Question 10

Mystic Pizza 
Mystic Pizza in Mystic,Connecticut,advertises that they deliver your pizza within 15 minutes of placing an order or it is free.A sample of 25 customers is selected at random.The average delivery time in the sample was 13 minutes with a sample standard deviation of 4 minutes. ​ ​
-{Mystic Pizza Narrative} We want to know whether Mystic can make this claim or not.Test to determine if we can infer at the 5% significance level that the population mean delivery time is less than 15 minutes.

Accepted Answer

The answer of Mystic Pizza 
Mystic Pizza in Mystic,Connecticut,advertises that...

Question 11

The air pumps at service stations come equipped with a gauge to regulate the air pressure of tires.A mechanic believes that the gauges are in error by at least 3 pounds per square inch.To test his belief he takes a random example of 50 air pump gauges and determines the difference between the true pressure (as measured by an accurate measuring device)and the pressure shown on the air pump gauge.The mean and the standard deviation of the sample are  = 3.4 and s = 1.2.Can the mechanic infer that he is correct at the 5% significance level? Assume tire pressures have a normal distribution.

Accepted Answer

The answer of The air pumps at service stations come...

Question 12

The chi-squared distribution can be used in constructing confidence intervals and carrying out hypothesis tests regarding the value of a population variance.

Accepted Answer

The chi-squared distribution is commonly used in statistical inference for testing hypotheses and constructing confidence intervals related to population variances. Therefore, option A is correct.

Question 13

Tire Rotation
 The manager of a service station is in the process of analyzing the number of times car owners rotate the tires on their cars.She believes that the average motorist rotates his or her car's tires less frequently than recommended by the owner's manual (two times per year).In a preliminary survey she asked 14 car owners how many times they rotated their cars' tires in the last 12 months.The results are 1,1,2,0,3,3,0,1,0,1,2,3,3,and 1. ​ ​
-{Tire Rotation Narrative} What condition is required in order to analyze this data using a t-test?

Accepted Answer

The answer of Tire Rotation
 The manager of a service...

Question 14

A random sample of 25 observations is selected from a normally distributed population.The sample variance is 10.In the 95% confidence interval for the population variance,the upper limit is:
A)19.353
B)17.331
C)17.110
D)6.097

Accepted Answer

The upper limit of the 95% confidence interval for the population variance is calculated using the chi-square distribution. For a sample size of 25, the degrees of freedom is 24. The upper critical value for chi-square at 0.025 significance level (for a two-tailed test) is approximately 39.364. The formula for the upper limit is (n-1)s^2/χ²(α/2), which gives (24*10)/39.364 = 19.353.

Question 15

A life insurance representative believes that the mean age of people who buy their first life insurance plan is less than 35.To test his belief he takes a random sample of 15 customers who have just purchased their first life insurance.Their ages are 42,43,28,34,30,36,25,29,32,33,27,30,22,37,and 40.There is not enough evidence to say the data are nonnormal.Can we conclude at the 1% significance level that the insurance representative is correct?

Accepted Answer

The answer of A life insurance representative believes that the...

Question 16

During a natural gas shortage,a gas company randomly sampled residential gas meters in order to monitor daily gas consumption.On a particular day,a sample of 100 meters showed a sample mean of 250 cubic feet and a sample standard deviation of 50 cubic feet.Provide a 90% confidence interval estimate of the mean gas consumption for the population.

Accepted Answer

The answer of During a natural gas shortage,a gas company...

Question 17

A condition of using the chi-squared test statistic for testing the variance is that the population random variable have a normal distribution.

Accepted Answer

The chi-squared test for variance assumes that the data are sampled from a population that follows a normal distribution, as the test statistic is derived under this condition.

Question 18

A random sample is drawn from a normal distribution with mean μ and variance σ².The random variable (n − 1)S² / σ² has a chi-squared probability distribution with n degrees of freedom.

Accepted Answer

The correct distribution has (n - 1) degrees of freedom, not n.

Question 19

The test statistic used to test hypotheses about the population variance is given by χ² = (n − 1)s² / σ²,which is chi-squared distributed with n − 1 degrees of freedom when the population random variable is normally distributed with variance equal to σ².

Accepted Answer

This is a correct statement about the test statistic used to test hypotheses about the population variance. The test statistic, denoted by χ², follows a chi-squared distribution with (n-1) degrees of freedom, where n is the sample size, when the population random variable is normally distributed with variance equal to σ².

Question 20

The statistic (n − 1)s² / σ² has a chi-squared distribution with n − 1 degrees of freedom if: A)the sample has a Student t-distribution with degrees of freedom equal to n − 1. B)the sample is normally distributed with variance equal to s². C)the population is normally distributed with variance equal to σ². D)All of these choices are true.

Accepted Answer

The given statistic is a ratio of the sample standard deviation to the population standard deviation, multiplied by a constant term. Therefore, it depends on the population variance (σ^2) and not on the sample size or the distribution of the sample. In addition, the numerator of the statistic follows a chi-squared distribution with n-1 degrees of freedom, which means that the denominator should be a constant multiple of σ^2 to result in a chi-squared distribution. Thus, the correct choice is C - the population is normally distributed with variance equal to σ^2.

Quiz 12: Inference About a Population

Which of the following is an example illustrating the use of variance?

Two uses for the population variance are to measure risk and consistency.

The area to the right of a certain critical value on a chi-squared distribution is 0.025.For 5 degrees of freedom,the critical value is 11.143.

A left-tailed area in the chi-squared distribution equals 0.95.For 6 degrees of freedom the critical value equals 12.592.

The statistic s² is an unbiased and consistent estimator of σ².

The chi-squared distribution can be used in constructing confidence intervals and carrying out hypothesis tests regarding the value of a population variance.

A random sample of 25 observations is selected from a normally distributed population.The sample variance is 10.In the 95% confidence interval for the population variance,the upper limit is:

A condition of using the chi-squared test statistic for testing the variance is that the population random variable have a normal distribution.

A random sample is drawn from a normal distribution with mean μ and variance σ².The random variable (n − 1)S² / σ² has a chi-squared probability distribution with n degrees of freedom.

The test statistic used to test hypotheses about the population variance is given by χ² = (n − 1)s² / σ²,which is chi-squared distributed with n − 1 degrees of freedom when the population random variable is normally distributed with variance equal to σ².

The statistic (n − 1) s² / σ² has a chi-squared distribution with n − 1 degrees of freedom if:

Quiz 12: Inference About a Population

Which of the following is an example illustrating the use of variance?

Two uses for the population variance are to measure risk and consistency.

The area to the right of a certain critical value on a chi-squared distribution is 0.025.For 5 degrees of freedom,the critical value is 11.143.

A left-tailed area in the chi-squared distribution equals 0.95.For 6 degrees of freedom the critical value equals 12.592.

The statistic s2 is an unbiased and consistent estimator of σ2.

The chi-squared distribution can be used in constructing confidence intervals and carrying out hypothesis tests regarding the value of a population variance.

A random sample of 25 observations is selected from a normally distributed population.The sample variance is 10.In the 95% confidence interval for the population variance,the upper limit is:

A condition of using the chi-squared test statistic for testing the variance is that the population random variable have a normal distribution.

A random sample is drawn from a normal distribution with mean μ and variance σ2.The random variable (n − 1)S2 / σ2 has a chi-squared probability distribution with n degrees of freedom.

The test statistic used to test hypotheses about the population variance is given by χ2 = (n − 1)s2 / σ2,which is chi-squared distributed with n − 1 degrees of freedom when the population random variable is normally distributed with variance equal to σ2.

The statistic (n − 1) s2 / σ2 has a chi-squared distribution with n − 1 degrees of freedom if:

The statistic s² is an unbiased and consistent estimator of σ².

A random sample is drawn from a normal distribution with mean μ and variance σ².The random variable (n − 1)S² / σ² has a chi-squared probability distribution with n degrees of freedom.

The test statistic used to test hypotheses about the population variance is given by χ² = (n − 1)s² / σ²,which is chi-squared distributed with n − 1 degrees of freedom when the population random variable is normally distributed with variance equal to σ².

The statistic (n − 1) s² / σ² has a chi-squared distribution with n − 1 degrees of freedom if: