Elementary Statistics

Robert Johnson, Patricia Kuby

Chapter 8

Introduction to Statistical Inferences - all with Video Answers

Educators

Section 1

The Nature of Estimation

Problem 1

A random sample of 50 female American health professionals yielded the following height data.
$$\begin{array}{llllllllll}
\hline 65 & 66 & 64 & 67 & 59 & 69 & 66 & 69 & 64 & 62 \\
63 & 62 & 63 & 64 & 72 & 66 & 65 & 64 & 67 & 68 \\
70 & 63 & 63 & 68 & 58 & 60 & 64 & 66 & 64 & 62 \\
65 & 69 & 64 & 69 & 62 & 58 & 66 & 68 & 59 & 56 \\
64 & 66 & 65 & 69 & 67 & 67 & 68 & 62 & 70 & 62 \\
\hline
\end{array}$$
a. What population was sampled to obtain the height data listed above?
b. Describe the sample data using the mean and standard deviation, plus any other numerical statistic(s) that help describe the sample.
c. Construct a histogram and comment on the shape of the distribution. Construct any other graph(s) that help describe the sample.
d. Using the statistics found in parts b and c, estimate the mean height of all American female health professionals using a single value. Using an interval.
e. What quality of the interval estimate would improve the worth of the interval?

Tyler T.

Tyler T.

Numerade Educator

Problem 2

Referring to Exercise 8.1:
a. How is the distribution of the sample height data on
p. 344 related to: (1) The distribution of the population? ( 2) The sampling distribution of sample means?
b. Using the techniques of Chapter 7 , find the limits that would bound the middle $90 \%$ of the sampling distribution of sample means for samples of size 50 selected randomly from the population of female heights with a known mean of 63.7 inches and a standard deviation of 2.75 inches.
c. On the histogram drawn in Exercise 8.1:(1) Draw a vertical line at the population mean of $63.7 .(2)$ Draw a horizontal line segment showing the interval found in part b. Does the sample mean found in Exercise 8.1 (b) fall in the interval? Answer yes or no, and explain what this means.
d. Using the techniques of Chapter $7,$ find $P(\bar{x} \geq 64.7)$ for
a random sample of 50 drawn from a population with a known mean of 63.7 inches and a standard deviation of 2.75 inches. Explain the meaning of the resulting value.
e. Does the sample of 50 height data values appear to belong to the population described by the NCHS? Explain.
f. Review the above answers and consider how the SDSM and CLT from Chapter 7 might be used to make an improvement in the interval estimate.

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Problem 3

Explain the difference between a point estimate and an interval estimate.

Tyler T.

Numerade Educator

Problem 4

Identify each numerical value by "name" (e.g., mean, variance and by symbol (e.g., $\bar{x}$ ):

a. The mean height of 24 junior high school girls is $4 \mathrm{ft}$
11 in.
b. The standard deviation for IQ scores is 16
c. The variance among the test scores on last week's exam was $190 .$
d. The mean height of all cadets who have ever entered West Point is 69 inches.

Tyler T.

Numerade Educator

Problem 5

A random sample of the amount paid (in dollars) for taxi fare from downtown to the airport was obtained:
$$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline 15 & 19 & 17 & 23 & 21 & 17 & 16 & 18 & 12 & 18 & 20 & 22 & 15 & 18 & 20 \\
\hline
\end{array}$$
Use the data to find a point estimate for each of the following parameters.
a. Mean
b. Variance
c. Standard deviation

Tyler T.

Numerade Educator

Problem 6

The number of engines owned per fire department was obtained from a random sample taken from the profiles of fire departments from across the United States (Firehouse/June 2003).
$$\begin{array}{lllllll}
\hline 29 & 8 & 7 & 33 & 21 & 26 & 0 & 11 & 4 & 54 & 7 & 4 \\
\hline
\end{array}$$
Use the data to find a point estimate for each of the following parameters.
a. Mean
b. Variance
c. Standard deviation

Tyler T.

Numerade Educator

Problem 7

In each diagram below, I and II represent sampling distributions of two statistics that might be used to estimate a parameter. In each case, identify the statistic that you think would be the better estimator, or neither, and describe why it is your choice.

Sara S.

Numerade Educator

Problem 8

Suppose that there are two statistics that will serve as an estimator for the same parameter. One of them is biased, and the other is unbiased.
a. Everything else being equal, explain why you usually would prefer an unbiased estimator to a biased estimator.
b. If a statistic is unbiased, does that ensure that it is a good estimator? Why or why not? What other considerations must be taken into account?
c. Describe a situation that might occur in which the biased statistic might be a better choice as an estimator than the unbiased statistic.

Tyler T.

Numerade Educator

Problem 9

The use of a tremendously large sample does not ensure quality in an estimator. What problems do you anticipate with very large samples?

Tyler T.

Numerade Educator

Problem 10

Being unbiased and having a small variability are two desirable characteristics of a statistic if it is going to be used as an estimator. Describe how the SDSM addresses both of these properties when estimating the mean of a population.

Sara S.

Numerade Educator

Problem 11

The U.S. Census Bureau reports that the estimated mean U.S. married-couple family income is $\$ 90,835 \pm \$ 101 .$ The Bureau describes the margin of error as providing a $90 \%$ probability that the interval defined by the estimate minus the margin of error and the estimate plus the margin of error (the lower and upper confidence bounds) contains the true value.
a. What is the population and variable of interest?
b. What parameter is being estimated? What is its estimated value?
c. How is the margin of error related to the maximum error of estimate?
d. What value is being reported as the margin of error?
e. What level of confidence is being reported?
f. Find the confidence interval and state exactly what
it represents.

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Problem 12

Consumer Reports National Research Center reported that $76 \%$ of women respond, "Daily or more often" when asked, "How often is your bed made?" As a footnote, this additional information is included: margin of error ±3.2 percentage points.
a. What is the population and variable of interest?
b. What parameter is being estimated? What is its estimated value?
c. What value is being reported as the margin of error?
d. Find the interval and state exactly what it represents.
e. What additional information might you want to know about this confidence interval?

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Problem 13

Explain why the standard error of sample means is 3 for the rivet example on page 343

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Problem 14

a. Verify that a $95 \%$ level of confidence requires a $1.96-$ standard-deviation interval.
b. $\quad$ Verify that the level of confidence for a 2-standard-deviation interval is $95.45 \%$

Sara S.

Numerade Educator

Problem 15

Find the level of confidence assigned to an interval estimate of the mean formed using the following intervals:
a. $\quad \bar{x}-1.28 \cdot \sigma_{\bar{x}}$ to $\bar{x}+1.28 \cdot \sigma_{\bar{x}}$
b. $\bar{x}-1.44 \cdot \sigma_{\bar{x}}$ to $\bar{x}+1.44 \cdot \sigma_{\bar{x}}$
c. $\quad \bar{x}-1.96 \cdot \sigma_{\bar{x}}$ to $\bar{x}+1.96 \cdot \sigma_{\bar{x}}$
d. $\bar{x}-2.33 \cdot \sigma_{\bar{x}}$ to $\bar{x}+2.33 \cdot \sigma_{\bar{x}}$

Sara S.

Numerade Educator

Problem 16

Find the level of confidence assigned to an interval estimate of the mean formed using the following intervals:
a. $\quad \bar{x}-1.15 \cdot \sigma_{\bar{x}}$ to $\bar{x}+1.15 \cdot \sigma_{\bar{x}}$
b. $\quad \bar{x}-1.65 \cdot \sigma_{x}$ to $\bar{x}+1.65 \cdot \sigma_{x}$
c. $\quad \bar{x}-2.17 \cdot \sigma_{\bar{x}}$ to $\bar{x}+2.17 \cdot \sigma_{\bar{x}}$
d. $\bar{x}-2.58 \cdot \sigma_{x}$ to $\bar{x}+2.58 \cdot \sigma_{x}$

Sara S.

Numerade Educator

Problem 17

Population requirement for primary hip-replacement surgery: a cross-sectional study" was conducted by the University of Bristol in the United Kingdom. The findings resulted in the following statement: "The prevalence of self reported hip pain was 107 per 1000 (95\% CI $101-113$ ) for men and 173 per $1000(166-180)$ for women."
a. Explain the meaning of the confidence interval, $95 \%$ CI $101-113$
b. Find the standard error for the men's self-reported hip pain $95 \%$ confidence interval.
c. Assuming the women's data was also from a $95 \%$ confidence interval, find the standard error.

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Problem 18

A sample of 25 of 174 funded projects revealed that 19 were valued at $\$ 17,320$ each and 6 were valued at $\$ 20,200$ each. From the sample data, estimate the total value of the funding for all the projects.

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Problem 19

Using the Old Faithful eruption information in Applied Example 8.1 on page 344 :
a. What does "3:19 PM ±10 min." mean? Explain.
b. Did this eruption occur during the predicted time interval?
c. What does "90\% of the time" mean? Explain.

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Problem 20

A recruiter estimates that if you are hired to work for her company and you put in a full week at the commissioned sales representative position she is offering, you will make "\$525 plus or minus $\$ 250,80 \%$ of the time." She adds, "It all depends on you!"
a. What does "\$525 plus or minus $\$ 250 "$ mean?
b. What does "80\% of the time" mean?
c. If you make $\$ 300$ to the nearest $\$ 10$ most weeks, will she have told you the truth? Explain.

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