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Statistics Learning From Data

Thomas H. Short, Roxy Peck

Chapter 8

Sampling Variability and Sampling Distributions - all with Video Answers

Educators

Section 1

Statistics and Sampling Variability

02:47

Problem 1

A random sample of 1000 students at a large college included 428 who had one or more credit cards. For this sample, $\hat{p}=\frac{428}{1000}=0.428 .$ If another random sample of 1000 students from this university were selected, would you expect that $\hat{p}$ for that sample would also be 0.428 ? Explain why or why not.

Bryan Luo
Bryan Luo
Numerade Educator
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Problem 2

Consider the two relative frequency histograms at the bottom of this page. The histogram on the left was constructed by selecting 100 different random samples of size 50 from a population consisting of $55 \%$ females and $45 \%$ males. For each sample, the sample proportion of females, $\hat{p},$ was calculated. The $100 \hat{p}$ values were used to construct the histogram. The histogram on the right was constructed in a similar way, except that the samples were of size 75 instead of 50 .
a. Which of the two histograms shows more sample-tosample variability? How can you tell?
b. For which of the two sample sizes, $n=50$ or $n=75,$ do you think the value of $\hat{p}$ is more likely to be close to $0.55 ?$ What about the given histograms supports your choice?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:22

Problem 3

Consider the following statement:
The Department of Motor Vehicles reports that the proportion of all vehicles registered in California that are imports is $0.22 .$
a. Is the number that appears in boldface in this statement a sample proportion or a population proportion?
b. Which of the following use of notation is correct, $p=0.22$ or $\hat{p}=0.22 ?$ (Hint: See definitions and notation on page $403 .$ )

Nick Johnson
Nick Johnson
Numerade Educator
01:43

Problem 4

Consider the following statement:
A sample of size 100 was selected from those admitted to a particular college in fall 2017 . The proportion of these 100 who were transfer students is $\mathbf{0} . \mathbf{3 8}$.
a. Is the number that appears in boldface in this statement a sample proportion or a population proportion?
b. Which of the following use of notation is correct, $p=0.38$ or $\hat{p}=0.38 ?$

James Kiss
James Kiss
Numerade Educator
00:55

Problem 5

Explain the difference between $p$ and $p$.

Natalie Johns
Natalie Johns
Numerade Educator
01:32

Problem 6

A random sample of 100 employees of a large company included 37 who had worked for the company for less than one year. For this sample, $\hat{p}=\frac{37}{100}=0.37 .$ If a different random sample of 100 employees were selected, would you expect that $\hat{p}$ for that sample would also be $0.37 ?$ Explain why or why not.

Kari Hasz
Kari Hasz
Numerade Educator
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Problem 7

Consider the two relative frequency histograms at the top of the next page. The histogram on the left was constructed by selecting 100 different random samples of size 40 from a population consisting of $20 \%$ part-time students and $80 \%$ full-time students. For each sample, the sample proportion of part-time students, $\hat{p},$ was calculated. The $100 \hat{p}$ values were used to construct the histogram. The histogram on the right was constructed in a similar way, but using samples of size 70 .
a. Which of the two histograms indicates that the value of $\hat{p}$ has smaller sample-to-sample variability? How can you tell?
b. For which of the two sample sizes, $n=40$ or $n=70,$ do you think the value of $\hat{p}$ would be less likely to be close to $0.20 ?$ What about the given histograms supports your choice?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:45

Problem 8

Explain what the term sampling variability means in the context of using a sample proportion to estimate a population proportion.

Nick Johnson
Nick Johnson
Numerade Educator
03:19

Problem 9

Consider the following statement:
Fifty people were selected at random from those attending a football game. The proportion of these 50 who made a food or beverage purchase while at the game was 0.83 .
a. Is the number that appears in boldface in this statement a sample proportion or a population proportion?
b. Which of the following use of notation is correct, $p=0.83$ or $\hat{p}=0.83 ?$

Nick Johnson
Nick Johnson
Numerade Educator
02:22

Problem 10

Consider the following statement:
The proportion of all calls made to a county $9-1-1$ emergency number during the year 2017 that were nonemergency calls was 0.14
a. Is the number that appears in boldface in this statement a sample proportion or a population proportion?
b. Which of the following use of notation is correct, $p=0.14$ or $\hat{p}=0.14 ?$

Nick Johnson
Nick Johnson
Numerade Educator
00:45

Problem 11

Explain what it means when we say the value of a sample statistic varies from sample to sample.

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 12

Consider the two relative frequency histograms at the top of the next page. The histogram on the left was constructed by selecting 100 different random samples of size 35 from a population in which $17 \%$ donated to a nonprofit organization. For each sample, the sample proportion $\hat{p}$ was calculated and then the $100 \hat{p}$ values were used to construct the histogram. The histogram on the right was constructed in a similar way but used samples of size 110 .
a. Which of the two histograms indicates that the value of $\hat{p}$ has more sample-to-sample variability? How can you tell?
b. For which of the two sample sizes, $n=35$ or $n=110,$ do you think the value of $\hat{p}$ would be more likely to be close to the actual population proportion of $p=0.17 ?$ What about the given histograms supports your choice?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:55

Problem 13

Consider the following statement:
In a sample of 20 passengers selected from those who flew from Dallas to New York City in April $2017,$ the proportion who checked luggage was $\mathbf{0} . \mathbf{4 5}$.
a. Is the number that appears in boldface in this statement a sample proportion or a population proportion?
b. Which of the following use of notation is correct, $p=0.45$ or $\hat{p}=0.45 ?$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:05

Problem 14

Consider the following statement:
A county tax assessor reported that the proportion of property owners who paid 2016 property taxes on time was $\mathbf{0} . \mathbf{9 3}$.
a. Is the number that appears in boldface in this statement a sample proportion or a population proportion?
b. Which of the following use of notation is correct, $p=0.93$ or $\hat{p}=0.93 ?$

Qudsiya Anis
Qudsiya Anis
Numerade Educator