Deck 8: Confidence Interval Estimation

Question 1

The following values have been calculated using the T DIST and T INV functions in Excel^®. These values come from a t- distribution with 15 degrees of freedom.
These values represent the probability to the right of the given positive values. These values represent the positive t- value for a given probability in both tails (sum of both tails).
What is the probability of a t-value smaller than 1.00?

Answer 1

0.8334

Question 2

When we replace with the sample standard deviation (s), we introduce a new source of variability and the sampling distribution becomes the:

A) t -distribution
B) F- distribution
C) chi-square distribution
D) normal distribution

Answer 2

t -distribution

Question 3

The following values have been calculated using the T DIST and T INV functions in Excel^®. These values come from a t- distribution with 15 degrees of freedom.
These values represent the probability to the right of the given positive values. These values represent the positive t- value for a given probability in both tails (sum of both tails).
What would be the t-value where 0.05 of the values are in the upper tail?

Answer 3

+ 1.7531

Question 4

If you are constructing a confidence interval for a single mean, the confidence interval will _____ with an increase in the sample size.

A) decrease
B) increase
C) stay the same
D) increase or decrease, depending on the sample data,

Question 5

The general form of a confidence interval is:

A) Point Estimate = Multiple × Standard Error
B) Point Estimate = Multiple +Standard Error
C) Point Estimate = Multiple - Standard Error
D) Point Estimate = Multiple ± Standard Error

Question 6

A parameter such as is sometimes referred to as a(n) ____ parameter, because many times, we need its value even though it is not the parameter of primary interest.

A) special
B) random
C) nuisance
D) independent
E) dependent

Question 7

The following values have been calculated using the T DIST and T INV functions in Excel^®. These values come from a t- distribution with 15 degrees of freedom.
These values represent the probability to the right of the given positive values. These values represent the positive t- value for a given probability in both tails (sum of both tails).
What would be the t-values where 0.10 of the values are in both tails (sum of both tails)?

Question 8

(A) Compute has a t-distribution with 15 degrees of freedom.
(B) Compute has a t-distribution with 150 degrees of freedom.
(C) How do you explain the difference between the results obtained in (A) and (B)?
(D) Compute where Z is a standard normal random variable.
(E) Compare the results of (D) to the results obtained in (A) and (B). How do you explain the difference in these probabilities?

Question 9

Two independent samples of sizes 50 and 50 are randomly selected from two populations to test the difference between the population means, . The sampling distribution of the sample mean difference is:

A) normally distributed
B) approximately normal
C) t - distributed with 98 degrees of freedom
D) chi-squared distributed with 99 degrees of freedom

Question 10

The t-distribution for developing a confidence interval for a mean has degrees of freedom equal to:

A) n + 2
B) n +1
C) n
D) n - 1
E) n - 2

Question 11

The following values have been calculated using the T DIST and T INV functions in Excel^®. These values come from a t- distribution with 15 degrees of freedom.
These values represent the probability to the right of the given positive values. These values represent the positive t- value for a given probability in both tails (sum of both tails).
What is the probability of a t-value larger than 1.20?

Question 12

The chi-square and F-distributions are used primarily to make inferences about population ____.

A) means
B) variances
C) medians
D) modes
E) proportions

Question 13

A confidence interval is an interval that, with a stated level of confidence, captures a population parameter.

Question 14

The approximate standard error of the point estimate of the population total is:

A)
B)
C)
D)

Question 15

The following values have been calculated using the T DIST and T INV functions in Excel^®. These values come from a t- distribution with 15 degrees of freedom.
These values represent the probability to the right of the given positive values. These values represent the positive t- value for a given probability in both tails (sum of both tails).
What is the probability of a t-value between -1.40 and +1.40?

Question 16

If you increase the confidence level, the confidence interval .

A) decreases
B) increases
C) stays the same
D) may increase or decrease, depending on the sample data

Question 17

When samples of size n are drawn from a population, then the sampling distribution of the sample mean is approximately normal, provided that n is reasonably large.

Question 18

The t-distribution and the standard normal distribution are practically indistinguishable as the degrees of freedom increase.

Question 19

The following values have been calculated using the T DIST and T INV functions in Excel^®. These values come from a t- distribution with 15 degrees of freedom.
These values represent the probability to the right of the given positive values. These values represent the positive t- value for a given probability in both tails (sum of both tails).
What would be the t-values where 0.95 of the values would fall within this interval?

Question 20

As the sample size increases, the t-distribution becomes more similar to the ____ distribution.

A) normal
B) exponential
C) multinominal
D) chi-square
E) binomial

Question 21

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if the sample size n is greater than 30.

Question 22

The average annual household income levels of citizens of selected U.S. cities are shown below.
(A) Use Excel^® to obtain a simple random sample of size 10 from this frame.
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(B) Using the sample generated in (A), construct a 95% confidence interval for the mean average annual household income level of citizens in the selected U.S. cities. Assume that the population consists of all average annual household income levels in the given frame.
​
(C) Interpret the 95% confidence interval constructed in (B).
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(D) Does the 95% confidence interval contain the actual population mean? If not, explain why not. What proportion of many similarly constructed confidence intervals should include the true population mean value?

Question 23

The 95% confidence interval for the population mean , given that the sample size n = 49 and the population standard deviation = 7, is .

Question 24

(A) Construct a 90% confidence interval estimate of the mean family dental expenses for all employees of this corporation.
(B) What assumption about the population distribution must be made to answer (A)?
(C) Interpret the 90% confidence interval constructed in (A).
(D) Suppose you used a 95% confidence interval in (A). What would be your answer?
(E) Suppose the fourth value were 593 instead of 93. What would be your answer to (A)? What effect does this change have on the confidence interval?
(F) Construct a 90% confidence interval estimate for the standard deviation of family dental expenses for all employees of this corporation.
(G) Interpret the 90% confidence interval constructed in (E).

Question 25

The standard error of the sampling distribution of the sample proportion , when the sample size n = 50 and the population proportion p = 0.25, is 0.00375.

Question 26

If the standard error of the sampling distribution of the sample proportion is 0.0324 for samples of size 200, then the population proportion must be 0.30.

Question 27

In order to construct a confidence interval estimate of the population mean , the value of must be given.

Question 28

In general, increasing the confidence level will narrow the confidence interval, and decreasing the confidence level widens the interval.

Question 29

(A) Construct a 95% confidence interval for the average width of an elevator rail. Do we need to assume that the width of elevator rails follows a normal distribution?
(B) How large a sample of elevator rails would we have to measure to ensure that we could estimate, with 95% confidence, the average diameter of an elevator rail within 0.01 inch?

Question 30

Suppose there are 500 accounts in a population. You sample 50 of them and find a sample mean of $500. What would be your estimate for the population total?

A) $5,000
B) $50,000
C) $250,000
D) $500,000
E) none of these choices

Question 31

If a sample has 20 observations and a 95% confidence estimate for is needed, the appropriate value of t-multiple is 2.093

Question 32

The interval estimate 18.5 2.5 is developed for a population mean in which the sample standard deviation s is 7.5. Had s equaled 15 instead, the interval estimate would be 37 5.0.

Question 33

If you are constructing a confidence interval for a single mean, the confidence interval will with a decrease in the sample size.

A) decrease
B) increase
C) stay the same
D) increase or decrease, depending on the sample data

Question 34

We can form a confidence interval for the population total T by finding a confidence interval for the population mean in the usual way, and then multiplying the lower and upper limits the confidence interval by the population size N.

Question 35

(A) Construct a 95% confidence interval for the mean of the average annual credit account balances.
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(B) Interpret the 95% confidence interval constructed in (A).
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(C) Use the confidence interval constructed for (A) to help the store evaluate its criteria for whether or not the credit card program is worthwhile.

Question 36

The approximate standard error of the point estimate of the population total is .

Question 37

(A) Construct a 95% confidence interval for the mean playing time of all Willie Nelson recordings.
(B) Interpret the confidence interval you constructed in (A).

Question 38

A 90% confidence interval estimate for a population mean is determined to be 72.8 to 79.6. If the confidence level is reduced to 80%, the confidence interval for becomes narrower.

Question 39

(A) Construct a 90% confidence interval for the total value of all savings account balances within this bank. Assume that the population consists of all savings account balances in the frame.
(B) Interpret the 90% confidence interval constructed in (A).

Question 40

If you decrease the confidence level, the confidence interval:

A) decreases
B) increases
C) stays the same
D) may increase or decrease, depending on the sample data

Question 41

The lower limit of the 95% confidence interval for the population proportion p, given that n = 300; and = 0.10 is 0.1339.

Question 42

The shape of a chi-square distribution:

A) is symmetric
B) is skewed to the left
C) is skewed to the right
D) depends on the sample data

Question 43

If two random samples of sizes 30 and 35 are selected independently from two populations whose means are 85 and 90, then the mean of the sampling distribution of the sample mean difference, , equals 5.

Question 44

(A) Construct a 99% confidence interval for the proportion of company employees who prefer plan A. Assume that the population consists of the preferences of all employees in the frame.
(B) Interpret the 99% confidence interval constructed in (A).

Question 45

(A) Construct a 99% confidence interval for the standard deviation of the number of hours this firm's employees spend on work-related activities in a typical week.
(B) Interpret the 99% confidence interval constructed in (A).
(C) Given the target range of 40 to 60 hours of work per week, should senior management be concerned about the number of hours their employees are currently devoting to work? Explain why or why not.

Question 46

In general, the paired-sample procedure is appropriate when the samples are naturally paired in some way and there is a reasonably large positive correlation between the pairs. In this case, the paired-sample procedure makes more efficient use of the data and generally results in narrower confidence intervals.

Question 47

The upper limit of the 90% confidence interval for the population proportion p, given that n = 100; and = 0.20 is 0.2658.

Question 48

Two independent samples of sizes 20 and 25 are randomly selected from two normal populations with equal variances. In order to test the difference between the population means, the test statistic is:

A) a standard normal random variable
B) an approximately standard normal random variable
C) t-distributed with 45 degrees of freedom
D) t-distributed with 43 degrees of freedom

Question 49

The mean of the sampling distribution of the sample proportion , when the sample size n = 100 and the population proportion p = 0.15, is 15.0.

Question 50

In developing a confidence interval for the difference between two population means using two independent samples, we use the pooled estimate in estimating the standard error of the sampling distribution of the sample mean difference if the populations are normal with equal variances.

Question 51

If two random samples of size 40 each are selected independently from two populations whose variances are 35 and 45, then the standard error of the sampling distribution of the sample mean difference, , equals 1.4142.

Question 52

You are told that a random sample of 150 people from Iowa has been given cholesterol tests, and 60 of these people had levels over the "safe" count of 200. Construct a 95% confidence interval for the population proportion of people in Iowa with cholesterol levels over 200.

Question 53

(A) Construct a 95% confidence interval estimate of the population proportion of all customers who still own the cars they purchased six years ago.
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(B) How can the result in (A) be used by the automobile dealer to study satisfaction with cars purchased at the dealership?

Question 54

What is an example of a problem in which the sample data is likely to be paired?

A) the difference between the means of appraised and sales house prices
B) the difference between the proportion of defective items from two suppliers
C) the difference in the mean life of two major brands of batteries
D) the difference in the mean salaries for graduates in two different academic fields at a university
E) none of these choices

Question 55

The number of degrees of freedom needed to construct 90% confidence interval for the difference between means when the data are gathered from paired samples, with 15 observations in each sample, is:

A) 30
B) 15
C) 28
D) 14

Question 56

The degrees of freedom for the t and chi-square distributions is a numerical parameter of the distribution that defines the precise shape of the distribution.

Question 57

The confidence interval for the population standard deviation σ is centered at the point estimate, the sample standard deviation s.

Question 58

When the samples we want to compare pair in some natural way, such as a pretest/posttest for each person or husband/wife pairs, a more appropriate form of analysis is to not compare two separate variables, but instead to compare their ____.

A) difference
B) sum
C) ratio
D) total
E) product

Question 59

If a random sample of size 250 is taken from a population, where it is known that the population proportion p = 0.4, then the mean of the sampling distribution of the sample proportion is 0.60.

Question 60

In developing a confidence interval for the population standard deviation , we make use of the fact that the sampling distribution of the sample standard deviation s is not the normal distribution or the t-distribution, but rather a right-skewed distribution called the chi-square distribution, which (for this procedure) has n - 1 degrees of freedom.

Question 61

(A) Find a 95% confidence interval for the mean account balance on this store's credit card (the t-multiple with 39 degrees of freedom is 2.0227).
(B) What sample size would be needed to ensure that we could estimate the true mean account balance and have only 5 chances in 100 of being off by more than $100?

Question 62

Q-Mart is interested in comparing its male and female customers. Q-Mart would like to know if its female charge customers spend more money, on average, than its male charge customers. They have collected random samples of 25 female customers and 22 male customers. On average, women charge customers spend $102.23 and men charge customers spend $86.46. Some information are shown below.
(A) Use a t - value of 2.014 to calculate a 95% confidence interval for the difference between the average female purchase and the average male purchase. Would you conclude that there is a significant difference between females and males in this case? Explain.
(B) What are the degrees of freedom for the t-multiple in this calculation? Explain how you would calculate the degrees of freedom in this case.
(C) What is the assumption in this case that allows you to use the pooled standard deviation for this confidence interval?

Question 63

After calculating the sample size needed to estimate a population proportion to within 0.05, you have been told that the maximum allowable error (B) must be reduced to just 0.025. If the original calculation led to a sample size of 1000, the sample size will now have to be:

A) 2000
B) 4000
C) 1000
D) 8000

Question 64

You would like to estimate the average amount a family spends on food during a year. In the past, the standard deviation of the amount a family has spent on food during a year has been approximately $800. If you want to be 95% sure that you estimated average family food expenditures within $50, how many families do you need to survey?

Question 65

If we cannot make the strong assumption that the variances of two samples are equal, then we must use the pooled standard deviation in calculating the standard error of a difference between the means.

Question 66

Q-Mart is interested in comparing customers who use its own charge card with those who use other types of credit cards. Q-Mart would like to know if customers who use the Q-Mart card spend more money per visit, on average, than customers who use some other type of credit cards. They have collected information on a random sample of 38 charge customers as shown below. On average, the person using a Q-Mart card spends $192.81 per visit and customers using another type of card spend $104.47 per visit.
(A) Using a t - value of 2.0281, calculate a 95% confidence interval for the difference between the average Q-Mart charge and the average charge on another type of credit card.
(B) What are the degrees of freedom for the t - multiple in this calculation? Explain how you would calculate the degrees of freedom in this case.
(C) What is the assumption in this case that allows you to use the pooled standard deviation for this confidence interval?
(D) Would you conclude that there is a significant difference between the two types of customers in this case? Explain.

Question 67

In determining the sample size n for estimating the population proportion p, a conservative value of n can be obtained by using 0.50 as an estimate of p.

Question 68

In constructing a confidence interval estimate for the difference between the means of two populations, where the unknown population variances are assumed not to be equal, summary statistics computed from two independent samples are as follows: , , , , , and . Construct a 90% confidence interval for .

Question 69

A real estate agent has collected a random sample of 40 houses that were recently sold in Grand Rapids, Michigan. She is interested in comparing the appraised value and recent selling price (in thousands of dollars) of the houses in this particular market. The values of these two variables for each of the 40 randomly selected houses are shown below.
(A) Use the sample data to generate a 95% confidence interval for the mean difference between the appraised values and selling prices of the houses sold in Grand Rapids.
(B) Interpret the constructed confidence interval fin (A) for the real estate agent.

Question 70

(A) Construct a 95% confidence interval for the difference between the proportions of online and classroom customers who pass the final exam.
(B) Interpret the confidence interval obtained in (A).

Question 71

A company employs two shifts of workers. Each shift produces a type of gasket where the thickness is the critical dimension. The average thickness and the standard deviation of thickness for shift 1, based on a random sample of 40 gaskets, are 10.85 mm and 0.16 mm, respectively. The similar figures for shift 2, based on a random sample of 30 gaskets, are 10.90 mm and 0.19 mm. Let be the difference in thickness between shifts 1 and 2, and assume that the population variances are equal.
(A) Construct a 95% confidence interval for .
(B) Based on your answer to (A), are you convinced that the gaskets from shift 2 are, on average, wider than those from shift 1? Why or why not?

Question 72

(A) Determine a 95% confidence interval for the proportion defective for the process today.
(B) Based on your answer to (A), is it still reasonable to think the overall proportion defective produced by today's process is actually the targeted 4%? Explain your reasoning.
(C) The confidence interval in (A) is based on the assumption of a large sample size. Is this sample size sufficiently large in this example? Explain how you arrived at your answer.
(D) How many units would have to be sampled to be 95% confident that you can estimate the fraction of defective parts within 2% (using the information from today's sample)?

Question 73

Confidence intervals are a function of the:

A) population, the sample, and the standard deviation
B) sample, the variable of interest, and the degrees of freedom
C) data in the sample, the confidence level, and the sample size
D) sampling distribution, the confidence level, and the degrees of freedom
E) mean, median, and mode

Question 74

Samples of exam scores for employees before and after a training class are examples of paired data.

Question 75

You are attempting to estimate the average amount a family spends on food during a year. In the past, the standard deviation of the amount a family has spent on food during a year has been approximately $1200. If you want to be 99% sure that you have estimated average family food expenditures within $60, how many families do you need to survey?

Question 76

When you calculate the sample size for a proportion, you use an estimate for the population proportion; namely . A conservative value for n can be obtained by using = :

A) 0.01
B) 0.05
C) 0.10
D) 0.50
E) 1.00

Question 77

(A) You can be 95% confident that the mean salary for all production managers with at least 15 years of experience is between what two numbers (the t-multiple with 8 degrees of freedom is 2.306). What assumption are you making about the distribution of salaries?
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(B) What sample size would be needed to ensure that we could estimate the true mean salary of all production managers with more than 15 years of experience and have only 5 chances in 100 of being off by more than $4200?

Question 78

A market research consultant hired by Coke Classic Company is interested in estimating the difference between the proportions of female and male customers who favor Coke Classic over Pepsi Cola in Chicago. A random sample of 200 consumers from the market under investigation shows the following frequency distribution. ​
(A) Construct a 95% confidence interval for the difference between the proportions of male and female customers who prefer Coke Classic^® over Pepsi Cola^®.
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(B) Interpret the constructed confidence interval.

Question 79

For a given confidence level, the procedure for controlling interval length usually begins with the specification of the:

A) point estimate
B) population standard deviation, s
C) sample standard deviation, s
D) interval half-length, B

Question 80

If two samples contain the same number of observations, then the data must be paired.