Deck 11: Inference About a Population

Question 1

If the sampled population is nonnormal, the t -test of the population mean m is still valid, provided that the condition is not extreme.

Answer 1

True

Question 2

The statistic s² is an unbiased and consistent estimator of s ² .

Answer 2

True

Question 3

The t- distribution is used in a confidence interval for a mean when the actual standard error is not known.

Answer 3

True

Question 4

In forming a 95% confidence interval for a population mean from a sample size of 20, the number of degrees of freedom from the t -distribution equals 20.

Question 5

In estimating the population mean with the population standard deviation unknown, if the sample size is 16, there are 8 degrees of freedom.

Question 6

If a sample has 15 observations and a 95% confidence estimate for m is needed, the appropriate value of t is 1.753.

Question 7

In order to determine the p -value associated with hypothesis testing about the population mean m , it is necessary to know the value of the test statistic.

Question 8

The t- distribution assumes that the population is normally distributed.

Question 9

The statistic when the sampled population is normal is Student t -distributed with n degrees of freedom.

Question 10

The t -distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

Question 11

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

Question 12

The test statistic used to test hypotheses about the population variance is given by c ² = ( n - 1)s ² \ s ², which is chi-squared distributed with n - 1 degrees of freedom when the population is normally distributed with variance equal to s ² .

Question 13

A race car driver tested his car for time from 0 to 60 mph, and in 20 tests obtained an average of 48.5 seconds with a standard deviation of 1.47 seconds. A 95% confidence interval for the 0 to 60 time is 45.2 seconds to 51.8 seconds.

Question 14

If a sample has 18 observations and a 90% confidence estimate for m is needed, the appropriate value of t is 1.740.

Question 15

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.

Question 16

A random sample is drawn from a normal distribution with mean m and variance s ² . The random variable ( n - 1)S² \ s ² has a chi-squared probability distribution with n degrees of freedom.

Question 17

In order to interpret the p -value associated with hypothesis testing about the population mean m , it is necessary to know the value of the test statistic.

Question 18

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

Question 19

The t- distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known, regardless of the shape of the distribution in the population.

Question 20

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

Question 21

The statistic ( n - 1)s ² \ s ² 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 s ².
C)the population is normally distributed with variance equal to s ².
D)All of these choices are true.

Question 22

Based on sample data, the 90% confidence interval limits for the population mean are LCL = 170.86 and UCL = 195.42. If the 10% level of significance were used in testing the hypotheses H ₀: m = 201 vs. H ₁: m ¹ 201, the null hypothesis:

A)would be rejected.
B)would be accepted.
C)would fail to be rejected.
D)would become H ₀: m ¹ 201

Question 23

A robust estimator is one that is:

A)unbiased and symmetrical about zero.
B)consistent and is also mound-shaped.
C)efficient and less spread out.
D)not sensitive to moderate nonnormality.

Question 24

The t -test for a population mean is ____________________, meaning that if the population is nonnormal, the results of the test and confidence interval are still valid as long as the nonnormality is not extreme.

Question 25

When the population standard deviation is unknown and the population is normal, the test statistic for testing hypotheses about m is the ____________________- distribution with ____________________ degrees of freedom.

Question 26

A major electronics store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: = $50.50 and s ² = 400. A 95% confidence interval for the average amount the credit card customers spent on their first visit to the chain's new store in the mall is:

A)$50.50 ± $9.09.
B)$50.50 ± $10.12.
C)$50.50 ± $11.08.
D)None of these choices.

Question 27

For statistical inference about the mean of a single population when the population standard deviation is unknown, the degrees for freedom for the t- distribution equal n - 1 because we lose one degree of freedom by using the:

A)sample mean as an estimate of the population mean.
B)sample standard deviation as an estimate of the population standard deviation.
C)sample proportion as an estimate of the population proportion.
D)sample size as an estimate of the population size.

Question 28

Researchers determined that 60 Puffs tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Puffs users yielded the following data on the number of tissues used during a cold: = 52 and s = 22. Suppose the test statistic does not fall in the rejection region at a = 0.05. Which of the following conclusions is correct?

A)At a = 0.05, we do not reject H ₀.
B)At a = 0.05, we reject H ₀.
C)At a = 0.05, we accept H ₀.
D)Both a and c.

Question 29

A random sample of size 15 taken from a normally distributed population revealed a sample mean of 75 and a sample variance of 25. The upper limit of a 95% confidence interval for the population mean would equal:

A)77.77
B)72.23
C)88.85
D)77.27

Question 30

When the population standard deviation is ____________________ and the population is normal, the test statistic for testing hypotheses about m is the t -distribution with n - 1 degrees of freedom.

Question 31

When the population standard deviation is unknown and the population is ____________________, the test statistic for testing hypotheses about m is the t -distribution with n - 1 degrees of freedom.

Question 32

Researchers determined that 60 Puffs tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Puffs users yielded the following data on the number of tissues used during a cold: = 52 and s = 22. Suppose the alternative we wanted to test was H ₁: m a = 0.05 is:

A)reject H ₀ if t > 1.6604.
B)reject H ₀ if t - 1.6604.
C)reject H ₀ if t > 1.9842 or Z - 1.9842.
D)reject H ₀ if t - 1.9842.

Question 33

Which of the following conditions is needed regarding the chi-squared test statistic for the test of variance?

A)The population random variable must be normal.
B)The test statistic must be a non-negative number.
C)The test statistic must have a chi-squared distribution with n - 1 degrees of freedom.
D)All of these choices are true.

Question 34

Which of the following conditions is needed regarding the chi-squared test statistic for the test of variance?

A)The population random variable must be normal.
B)The test statistic must be a non-negative number.
C)The test statistic must have a chi-squared distribution with n - 1 degrees of freedom.
D)All of these choices are true.

Question 35

For a 99% confidence interval of the population mean based on a sample of n = 25 with s = 0.05, the critical value of t is:

A)2.7969
B)2.7874
C)2.4922
D)2.4851

Question 36

Which of the following is not 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.

Question 37

When a population is small, we must adjust the test statistic and interval estimator using the ____________________ population correction factor.

Question 38

The degrees of freedom for the test statistic for m when s is unknown is:

A)1
B)n
C)n - 1
D)None of these choices.

Question 39

Researchers determine that 60 Puffs tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Puffs users yielded the following data on the number of tissues used during a cold: = 52 and s = 22. Using the sample information provided, the value of the test statistic is:

A)t = (52 - 60)\ 22
B)t = (52 - 60)\ (22 \ 100)
C)t = (52 - 60)\ (22 \ 100²)
D)t = (52 - 60)\ (22 \ 10)

Question 40

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

Question 41

If we are testing for an improvement in the consistency of an operation or the manufacturing of a product, the alternative hypothesis would have a(n)____________________ sign in it.

Question 42

The test statistic to test hypotheses about the population variance has a chi-squared distribution with n - 1 degrees of freedom when the population random variable has a(n)____________________ distribution.

Question 43

The sum of squared deviations from the mean divided by the population variance has a(n)____________________ distribution with ____________________ degrees of freedom.

Question 44

The ____________________ confidence limit of the confidence interval estimator of the population variance is .

Question 45

Hourly Fees A random sample of 15 hourly fees for car washers (including tips)was drawn from a normal population. The sample mean and sample standard deviation were = $14.9 and s = $6.75. {Hourly Fees Narrative} Can we infer at the 5% significance level that the population mean is greater than 12, assuming that you know the population standard deviation is equal to 6.75?

Question 46

The test statistic used to test hypotheses about the population variance has a chi-squared distribution with ____________________ degrees of freedom.

Question 47

The ____________________ confidence limit of the confidence interval estimator of the population variance is .

Question 48

Hourly Fees A random sample of 15 hourly fees for car washers (including tips)was drawn from a normal population. The sample mean and sample standard deviation were = $14.9 and s = $6.75. {Hourly Fees Narrative} Can we infer at the 5% significance level that the mean fee for car washers (including tips)is greater than 12?

Question 49

Because of the greater uncertainty, the t -statistic will display greater ____________________ than the z -statistic.

Question 50

The t -statistic has two variables: the sample ____________________ and the sample ____________________.

Question 51

Single Mothers' Ages A random sample of 10 single mothers was drawn from a Obstetrics Clinic. Their ages are 22, 17, 27, 20, 23, 19, 24, 18, 19, and 24 years. {Single Mothers' Ages Narrative} Estimate the population mean with 90% confidence.

Question 52

____________________ populations allow us to use the confidence interval estimate of a mean to produce a confidence interval estimate of the population total.

Question 53

Single Mothers' Ages A random sample of 10 single mothers was drawn from a Obstetrics Clinic. Their ages are 22, 17, 27, 20, 23, 19, 24, 18, 19, and 24 years. {Single Mothers' Ages Narrative} Test to determine if we can infer at the 5% significance level that the population mean is not equal to 20.

Question 54

The chi-squared test and estimator of the population variance ____________________(are\are not)valid if the population is slightly to moderately nonnormal.

Question 55

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?

Question 56

Concert Tickets A simple random sample of 100 concert tickets was drawn from a normal population. The mean and standard deviation of the sample were $120 and $25, respectively. {Concert Tickets Narrative} Explain how to use the confidence interval to test the hypotheses at a = 0.10.

Question 57

Single Mothers' Ages A random sample of 10 single mothers was drawn from a Obstetrics Clinic. Their ages are 22, 17, 27, 20, 23, 19, 24, 18, 19, and 24 years. {Single Mothers' Ages Narrative} What is the required condition of the techniques used in the previous questions? What graphical device can you use to check to see if that required condition is satisfied?

Question 58

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} Does this data provide sufficient evidence at the 10% significance level to indicate that the manager is correct?

Question 59

Concert Tickets A simple random sample of 100 concert tickets was drawn from a normal population. The mean and standard deviation of the sample were $120 and $25, respectively. {Concert Tickets Narrative} Test the hypothesis H ₀: m = 125 vs. H ₁: m ¹ 125 at the 10% significance level.

Question 60

Concert Tickets A simple random sample of 100 concert tickets was drawn from a normal population. The mean and standard deviation of the sample were $120 and $25, respectively. {Concert Tickets Narrative} Estimate the population mean with 90% confidence.

Question 61

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.

Question 62

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.

Question 63

Science Quiz Scores Consider the hypotheses H ₀: s ² = 20 vs. H ₁: s ² > 20. Assume that Science scores on a 25-point quiz for a random sample of 5 students were drawn from a normal population. These were: 18, 16, 10, 13, and 23. {Science Quiz Scores Narrative} Estimate the population variance with 90% confidence.

Question 64

Science Quiz Scores Consider the hypotheses H ₀: s ² = 20 vs. H ₁: s ² > 20. Assume that Science scores on a 25-point quiz for a random sample of 5 students were drawn from a normal population. These were: 18, 16, 10, 13, and 23. {Science Quiz Scores Narrative} Test the hypotheses at the 10% significance level.

Question 65

A pharmaceutical company has just developed a new capsule to alleviate the symptoms of allergies and colds. However, they are concerned about the variability in the amount of time until the drug becomes effective. In a random sample of 10 individuals who suffer from allergies, the amount of time (in hours)for the capsule to take effect was recorded and listed as follows: 5, 7, 6, 10, 9, 12, 8, 17, 4, and 16. Estimate with 90% confidence the variance of the time for the drug to become effective.

Question 66

Applicants' Grades The grades of a sample of 10 applicants, selected at random from a large population, are 71, 86, 75, 63, 92, 70, 81, 59, 80, and 90. {Applicants' Grades Narrative} Construct a 90% confidence interval estimate for the population variance.

Question 67

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?

Question 68

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?

Question 69

Applicants' Grades The grades of a sample of 10 applicants, selected at random from a large population, are 71, 86, 75, 63, 92, 70, 81, 59, 80, and 90. {Applicants' Grades Narrative} Compute the sample variance.

Question 70

An investor is concerned with the risk associated with a portfolio of stocks. He draws a random sample of nine monthly returns (expressed as a percentage of the initial investment). These data follow: 2, 5, - 6, 10, 1, 2, - 3, 0, and 7. Find a 95% confidence interval estimate of the population variance.

Question 71

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.

Question 72

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.

Question 73

Applicants' Grades The grades of a sample of 10 applicants, selected at random from a large population, are 71, 86, 75, 63, 92, 70, 81, 59, 80, and 90. {Applicants' Grades Narrative} Construct a 90% confidence interval estimate for the population standard deviation.

Question 74

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?

Question 75

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.

Question 76

The sales manager of a large multinational corporation is concerned that some salespersons perform very well and others quite poorly. To help analyze the problem he draws a random sample of 20 salespersons, determines their commission incomes (in thousands of dollars), and calculates the following statistics: = $37.2, and s = $7.8. Do these statistics provide sufficient evidence at the 5% significance level to conclude that the population variance exceeds $35 million²?