Deck 8: Large-Sample Estimation

Question 1

A 95% lower confidence bound (LCB) for the population mean can be constructed using the following equation:
LCB =

Answer 1

False

Question 2

A quality control engineer wants to determine what proportion of defective parts are coming off the assembly line. Past experiments based on large sample sizes show this proportion to be 0.19. What sample size does the engineer need in order to estimate this proportion with a margin of error of 0.12 with 90% confidence?
______________

Answer 2

29

Question 3

If you wish to construct 92% lower confidence bound (LCB) for the difference between population means in case of large samples, then the z-value you should use is approximately:

A) 2.58
B) 1.96
C) 1.65
D) 1.41
E) 1.06

Answer 3

1.41

Question 4

A 90% upper confidence bound (UCB) for the population proportion p can be constructed using the following equation:
UCB =

Question 5

If you wish to construct 85% upper confidence bound (UCB) for the population proportion p, then the z-value you should use is approximately:

A) 2.58
B) 2.33
C) 1.96
D) 1.65
E) 1.04

Question 6

A state job service employee wishes to estimate the mean number of people who register with the service each week. How many weeks should be sampled in order to estimate the mean number of weekly registrants? The employee would like the margin of error to be less than 0.5 with probability of 0.95. Past records show the weekly standard deviation to be 2.5.
______________ weeks

Question 7

The Postmaster at the Harrington Post Office would like to compare the delivery times to two different locations which are the same number of miles from Harrington. A random sample of letters are to be divided into two equal groups, the first delivered to Location A and the second delivered to Location B. Each letter will be delivered on a randomly selected day and the number of days for each letter to arrive at its destination is recorded. The measurements for both groups are expected to have a range (variability) of approximately 4 days. If the estimate of the difference in mean delivery times is desired to be correct to within 1 day with probability equal to 0.99, how many letters must be included in each group? Assume n₁ = n₂ = n.
______________

Question 8

It is desired to compare the average ages at which men and women get their first license to drive. A random sample of 75 men yielded a mean and standard deviation of 17.3 and 4.7 years, respectively. A random sample of 96 women yielded a mean and standard deviation of 19.6 and 5.1 years, respectively. If it is desired to estimate the mean difference to within 1.5 years with 95% confidence, how large a sample should be taken from each population? Assume n₁ = n₂ = n.
______________

Question 9

Suppose you wish to estimate a population mean based on a sample of n observations. What sample size is required if you want your estimate to be within 2 of with probability equal to 0.95 if you know the population standard deviation is 12?

A) 239
B) 196
C) 139
D) 127
E) 98

Question 10

One-sided confidence bounds can be constructed for the population mean and population proportion p, but not for the difference between population means, or the difference between population proportions.

Question 11

A researcher wants to determine the proportion of elm trees in Superior, Wisconsin, dying of Dutch elm disease. Past experiments based on large sample sizes have shown this proportion to be 0.3. What sample size does the researcher need in order to estimate this proportion to within 0.04 with 95% confidence?
______________

Question 12

In a study of radio listening habits, a station owner would like to estimate the average number of hours that teenagers spend listening each day. If it is reasonable to assume that = 1.3 hours, how large a sample size is needed to be 90% confident that the sample mean is off by at most 30 minutes?
______________

Question 13

If you wish to construct 80% lower confidence bound (LCB) for the population mean , then the z-value you should use is approximately:

A) .84
B) 1.28
C) 1.96
D) 2.33
E) 2.58

Question 14

Suppose you wish to estimate a population proportion p based on sample of n observations. What sample size is required if you want your estimate to be within .03 of p with probability equal to 0.90?

A) 752
B) 423
C) 1,068
D) 1.529
E) none of these

Question 15

A laboratory technician is interested in the proportion of 1-liter containers used in the lab that are glass. How many containers should be sampled in order to estimate this proportion with a margin of error of less than 0.2 with 99% confidence?
______________

Question 16

A 99% upper confidence bound (UCB) for the difference between population means in case of large samples (both and greater than 30) can be constructed using the following equation:
UCB =

Question 17

A 95% lower confidence bound (LCB) for the difference between population proportions can be constructed using the following equation:
LCB =

Question 18

If you wish to construct 98% upper confidence bound (UCB) for the difference between population proportions, then the approximate z-value you should use is:

A) 2.33
B) 2.05
C) 1.96
D) 1.65
E) 1.41

Question 19

Trish attends an aerobics class four times each week. She would like to estimate the mean number of minutes of continuous exercise until her heart reaches 90 beats per minute. If it can be assumed that = 1.7 minutes, how large a sample is needed so that it will be possible to assert with 99% confidence that the sample mean has a margin of error of at most 0.62 minutes?
______________

Question 20

The sample size needed to estimate a population mean within 1.5 units with a 95% confidence when the population standard deviation equals 10 is:

A) 121
B) 13
C) 171
D) 54
E) 88

Question 21

Assume that two independent random samples of sizes and have been selected from binomial populations with parameters and , respectively. The standard error of the sampling distribution of , the difference between sample proportions, is estimated by .

Question 22

Assume that two independent random samples of sizes and have been selected from binomial populations with parameters and , respectively. The sampling distribution of , the difference between sample proportions, can be approximated by a normal distribution provided that , and are all greater than 5.

Question 23

A questionnaire is designed to investigate attitudes about political corruption in government. The experimenter would like to survey two different groups--Republicans and Democrats--and compare the responses to various "yes-no" questions for the two groups. The experimenter requires that the sampling error for the difference in the proportion of yes responses for the two groups is no more than 4 percentage points with probability equal to 0.95. If the two samples are both the same size, how large should the samples be?
______________

Question 24

If we wish to construct a 95% confidence interval estimate for the difference between two population proportions, then the confidence level would be:

A) 1.96
B) 0.475
C) 0.95
D) 0.05
E) 2.33

Question 25

If a 90% confidence interval estimate for the difference between two population proportions is to be constructed, then the confidence coefficient would be:

A) .10
B) .90
C) .05
D) .45
E) .35

Question 26

A process control engineer wishes to estimate the true proportion of defective computer chips with a margin of error of no more than 0.09 and with probability 0.90. How many observations does the engineer need to include in the sample to achieve his goal?
______________

Question 27

Two independent random samples of sizes and have been selected from binomial populations with parameters and , respectively, and resulted in 38 and 65 success, respectively. Then the standard error of is estimated as .077.

Question 28

A manufacturer wishes to estimate the mean time a battery pack will function before needing to be recharged with a margin of error of no more than 0.5 hours and with probability 0.95. If the standard deviation is known to be 1.5 hours, how many observations should be included in the sample?
______________

Question 29

You want to estimate the difference in grade point averages between two groups of college students accurate to within 0.20 grade point, with probability equal to 0.95. If the standard deviation of the grade point measurements is approximately equal to 0.5, how many students must be included in each group? (Assume that the groups will be of equal size.)
______________

Question 30

The mean of the sampling distribution of , the difference between sample proportions, is the difference between population proportions.

Question 31

The z-value needed to construct a 97.8% confidence interval estimate for the difference between two population proportions is:

A) 2.33
B) 2.29
C) 2.02
D) 1.96
E) 1.65

Question 32

A simple extension of the estimation of a binomial proportion p is the estimation of the difference between two binomial proportions and .

Question 33

Suppose you wish to estimate a population mean based on a random sample of n observations, and prior experience suggests that . If you wish to estimate correct to within 1.8, with probability equal to 0.95, how many observations should be included in your sample?
______________

Question 34

The z-value needed to construct a 92.5% confidence interval estimate for the difference between two population proportions is:

A) 2.58
B) 2.33
C) 1.96
D) 1.78
E) 1.33

Question 35

An airline executive estimates that 25% of all flights arrive late. How many flights must we include in a simple random sample if we want to be 90% confident that the true population proportion of flights that arrive late lies within 0.01 of our sample proportion?
______________

Question 36

A childcare agency was interested in examining the amount that families pay per child per month for childcare outside the home. A random sample of 64 families was selected and the mean and standard deviation were computed to be $675 and $80, respectively.
Find a 95% upper confidence bound for the true average amount spent per child per month on childcare outside the home.
______________

Question 37

Two independent random samples of sizes and have been selected from binomial populations with parameters and , respectively, and resulted in 38 and 65 success, respectively. Then, the point estimation of the difference is -27.

Question 38

A machine produces aluminum tins used in packaging cheese. A random sample of 1000 tins was selected and 43 were found to be defective. Find a 95% upper confidence bound for the true proportion of defective tins produced by the machine.
______________

Question 39

The manufacturer of a particular battery pack for a laptop computer claims the battery pack can function for 8 hours, on the average, before having to be recharged. A random sample of 36 such battery packs was selected and tested. The mean and standard deviation were found to be 6 hours and 1.8 hours, respectively.
Find a 95% lower confidence bound for the true average time the battery pack can function before having to be recharged.
______________

Question 40

Assume that the population standard deviation of annual incomes of all Michigan residents is $2,500. How many individuals must we include in a simple random sample if we want to be 95% confident that the population mean incomes lies within $150 of our sample mean income?
______________

Question 41

A manufacturing plant has two assembly lines for producing glass bottles. The plant manager was concerned about whether the proportion of defective bottles differs between the two lines. Two independent random samples were selected and the following summary data computed: Find a 95% confidence interval for the true difference in proportion of defective bottles produced by the two assembly lines.
______________
Based on the interval above, can one conclude there is a difference in proportion of defective bottles produced by the two lines?
______________
Explain.
________________________________________________________

Question 42

If you wish to estimate the difference between two population means when the population variances are known, the 90% confidence interval estimate can be developed using which of the following critical values of z?

A) 2.33
B) 1.28
C) 1.96
D) 2.58
E) 1.645

Question 43

In developing a confidence interval estimate for the difference between two population means, an increase in the size of the samples will result in:

A) a wider confidence interval
B) a narrower confidence interval
C) a smaller critical value z
D) the same confidence interval
E) none of these

Question 44

Increasing the confidence level for a confidence interval estimate for the difference between two population means, with all other things held constant, will result in a wider confidence interval estimate.

Question 45

In estimating the difference between two population means, if a 90% confidence interval estimate includes zero, then we can conclude that there is a 90% chance that the difference between the two population means is zero.

Question 46

The difference between two sample means is an unbiased estimator of the difference between two population means .

Question 47

When two independent random samples of sizes and have been selected from populations with means and , and variances and , which of the following is true?

A) The mean ofis.
B) The standard error ofis.
C) The standard error ofis estimated as.
D) All of these.
E) None of these.

Question 48

Suppose a 90% confidence interval for the mean time it takes to serve a customer at a drive-in bank is 120 seconds to 220 seconds. At the 90% confidence level, there is not enough evidence to conclude that the mean service time is not 200 seconds.

Question 49

The best estimator of the difference between two population means is the difference between two sample means .

Question 50

Instead of paying to support welfare recipients, many Californians want them to find jobs; if necessary, they want the state to create public service jobs for those who cannot find jobs in private industry. In a survey of 800 registered voters, 400 Republicans and 400 Democrats, 75% of the Republicans and 90% of the Democrats favored the creation of public service jobs. Use a large-sample estimation procedure to compare the proportions of Republicans and Democrats who favor creating public service jobs in the population of registered voters in California.
The approximate 95% confidence interval is:
______________
Based on the interval above, is there a difference in the proportion of Republicans and Democrats who favor creating public service jobs in California?
______________
Explain.
________________________________________________________

Question 51

If you wish to estimate the difference between two population means when the population variances are known, the 99% confidence interval estimate can be developed using which of the following critical values of z?

A) 2.575
B) 1.645
C) 2.325
D) 1.275
E) 2.582

Question 52

Suppose a 95% confidence interval for the mean height of a 12-year-old male in the United States is 54 to 65 inches. In repeated sampling, 95% of the intervals constructed will contain the interval from 54 to 65 inches.

Question 53

Two samples are said to be independent if they are selected at different points in time.

Question 54

Suppose a 90% confidence interval for the mean time it takes to serve a customer at a drive-in bank is 120 seconds to 220 seconds. In repeated sampling 90% of the intervals constructed using the appropriate formula will contain the actual mean time.

Question 55

A stylist at The Hair Care Palace gathered data on the number of hair colorings given on Saturdays and on weekdays. Her results are listed below. Assume the two samples were independently taken from normal populations. Find the point estimate of p₁ - p₂.
______________
Find the margin of error.
______________
Estimate the difference in the true proportions with a 99% confidence interval.
______________
Interpret this interval.
________________________________________________________

Question 56

In a study of the relationship between birth order and college success, an investigator found that 140 in a sample of 200 college graduates were firstborn or only children. In a sample of 120 non-graduates of comparable age and socioeconomic background, the number of firstborn or only children was 66. Estimate the difference between the proportions of firstborn or only children in the two populations from which these samples were drawn. Use a 90% confidence interval and interpret your results.
The approximate 90% confidence interval is:
______________
Based on the interval above, is there a difference between the proportions of firstborn or only children in the two populations?
______________
Explain.
________________________________________________________

Question 57

The standard error of the sampling distribution of is given by the formula:
SE = .

Question 58

When two independent random samples of sizes and have been selected from populations with means and , and variances and , which of the following is true?

A) If the sampled populations are normally distributed, then the sampling distribution ofis exactly normal only whenandare both 30 or more.
B) If the sampled populations are normally distributed, then the sampling distribution ofis exactly normal regardless of the sizes ofand.
C) If the sampled populations are not normally distributed, then the sampling distribution ofis approximately normally distributed regardless of the sizes ofand.
D) If the sampled populations are not normally distributed, then the sampling distribution ofis approximately normally distributed only ifis 30 or more.
E) None of these.

Question 59

In constructing a 95% confidence interval estimate for the difference between the means of two normally distributed populations, where the unknown population variances are assumed not to be equal, summary statistics computed from two independent samples are: , , , , , and . The upper confidence limit is:

A) 18.78
B) 6.78
C) 5.76
D) 77.3
E) 89.3

Question 60

When two independent random samples of sizes and have been selected from populations with means and and variances and , the standard error of the sampling distribution of the difference between the two sample means, is found by taking the square root of the sum of the two population variances; namely, .

Question 61

An Internet server conducted a survey of 400 of its customers and found that the average amount of time spent online was 12.5 hours per week with a standard deviation of 5.4 hours.
What shape do you think the random variable x, the number of hours spent online, has?
______________
If the distribution of the original measurements is not normal, you can still use the standard normal distribution to construct a confidence interval for , the average online time for all users of this Internet server. Why?
________________________________________________________
Construct a 95% confidence interval for the average online time for all users of the particular Internet server.
______________
If the Internet server claimed that its users averaged 15 hours of use per week, would you agree or disagree?
______________
Explain.
________________________________________________________

Question 62

The sample proportion is an unbiased estimator of the population proportion p.

Question 63

When constructing a confidence interval for a population parameter, we generally set the confidence coefficient ( ) close to 0 (usually between 0 and 0.05) because it is the probability that the interval does not include the actual value of the population parameter.

Question 64

In the formula: , the refers to the area in the lower tail or upper tail of the sampling distribution of the sample mean.

Question 65

The meat department of a local supermarket packages ground beef using meat trays of two sizes: one designed to hold approximately 1.5 pounds of meat, and one that holds approximately 3 pounds. A random sample of 36 packages in the smaller meat trays produced weight measurements with an average of 1.51 pounds and a standard deviation of 0.20 pound.
Construct a 99% confidence interval for the average weight of all packages sold in the small meat trays by this supermarket chain.
______________
What does the phrase "99% confident" mean?
________________________________________________________
Suppose that the quality control department of this supermarket chain intends that the amount of ground beef in the smaller trays should be 1.5 pound on average. Should the confidence interval in part (a) concern the quality control department?
______________
Explain.
________________________________________________________

Question 66

A computer laboratory manager was in charge of purchasing new battery packs for her lab of laptop computers. She narrowed her choices to two models that were available for her machines. Since the two models cost about the same, she was interested in determining whether there was a difference in the average time the battery packs would function before needing to be recharged. She took two independent random samples and computed the following summary information: Find a 95% confidence interval for the difference in average functioning time before recharging in the two models.
______________
Based on the interval above, can one conclude there is a difference in the true average functioning time before recharging between the two models of battery packs?
______________
Explain.
________________________________________________________

Question 67

A dieter believes that the average number of calories in a homemade peanut butter cookie is more than in a store-bought peanut butter cookie. The summary data are listed below. Estimate the difference in the mean calories between the two types of cookies using a 90% confidence interval.
______________

Question 68

A parent believes the average height for 14-year-old girls differs from that of 14-year-old boys. The summary data are listed below where height is in feet. Estimate the difference in the height between girls and boys using a 95% confidence interval.
______________
Based on your interval, do you think there is a significant difference between the true mean height of 14-year-old girls and boys?
______________
Explain.
________________________________________________________

Question 69

Suppose a 95% confidence interval for the mean height of a 12-year-old male in the United States is 54 to 65 inches. It can be said that 95% of 12-year-old males in the U.S. have height greater than or equal to 54 inches and less than or equal to 65 inches.

Question 70

A study was conducted to compare the mean numbers of police emergency calls per 8-hour shift in two districts of Los Angeles. Samples of 100 8-shifts were randomly selected from the police records for each of the two regions, and the number of emergency calls was recorded for each shift. The sample statistics are listed below: Find a 90% confidence interval for the difference in the mean numbers of police emergency calls per shift between the two districts of the city.
______________
Interpret the interval.
________________________________________________________

Question 71

An experiment was conducted to compare two diets A and B designed for weight reduction. Two groups of 50 overweight dieters each were randomly selected. One group was placed on diet A and the other on diet B, and their weight losses were recorded over a 30-day period. The means and standard deviations of the weight-loss measurements for the two groups are shown in the table. Find a 95% confidence interval for the difference in mean weight loss for the two diets.
______________
Interpret the interval.
________________________________________________________

Question 72

It is of interest to know if the average time it takes police to reach the scene of an accident differs from that of an ambulance to reach the same accident. Use the summary data listed below where time is in minutes. Estimate the difference in the times between the police and the ambulance using a 99% confidence interval.
______________
Does it appear that the times differ?
______________

Question 73

The lower and upper limits of the 68.26% confidence interval for the population mean , given that n = 49; = 75; and = 7, are 74 and 76, respectively.

Question 74

As the sample size increases and other factors are the same, the width of a confidence interval for a population mean tends to decrease.

Question 75

A comparison between the average jail time of bank robbers and car thieves yielded the following results (in years): Estimate ( ), the difference in mean years of jail time.
______________
Find the margin of error for your estimate.
______________

Question 76

The confidence coefficient is the probability that a confidence interval will enclose the estimated parameter.

Question 77

In order to construct a confidence interval estimate of the population proportion p, the value of p is needed.

Question 78

If the population variance is increased and other factors are the same, the width of a confidence interval for the population mean tends to increase.

Question 79

A 90% confidence interval estimate for a population mean is determined to be 62.8 to 73.4. If the confidence level is reduced to 80%, the confidence interval for becomes narrower.

Question 80

In developing an interval estimate for the population mean , the population standard deviation was assumed to be 6. The interval estimate was 45.0 1.5. Had equaled 12, the interval estimate would be 90 3.