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Statistics Learning From Data

Thomas H. Short, Roxy Peck

Chapter 9

Estimating a Population Proportion - all with Video Answers

Educators

Section 1

Selecting an Estimator

01:22

Problem 1

For estimating a population characteristic, why is an unbiased statistic with a small standard error preferred over an unbiased statistic with a larger standard error?

Sara Sasani
Sara Sasani
Numerade Educator
01:43

Problem 2

Three different statistics are being considered for estimating a population characteristic. The sampling distributions of the three statistics are shown in the following illustration:
Which of these statistics are unbiased estimators of the population characteristic?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:21

Problem 3

Three different statistics are being considered for estimating a population characteristic. The sampling distributions of the three statistics are shown in the following illustration:
Which statistic would you recommend? Explain your choice.

Sara Sasani
Sara Sasani
Numerade Educator
View

Problem 4

A researcher wants to estimate the proportion of students enrolled at a university who eat fast food more than three times in a typical week. Would the standard error of the sample proportion $\hat{p}$ be smaller for random samples of size $n=50$ or random samples of size $n=200 ?$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:57

Problem 5

Use the formula for the standard error of $\hat{p}$ to explain why
a. the standard error is greater when the value of the population proportion $p$ is near 0.5 than when it is near 1 .
b. the standard error of $\hat{p}$ is the same when the value of the population proportion is $p=0.2$ as it is when $p=0.8$.

Nick Johnson
Nick Johnson
Numerade Educator
01:47

Problem 6

A random sample will be selected from the population of all adult residents of a particular city. The sample proportion $\hat{p}$ will be used to estimate $p,$ the proportion of all adult residents who are employed full time. For which of the following situations will the estimate tend to be closest to the actual value of $p$ ?
i. $n=500, p=0.6$
ii. $n=450, p=0.7$
iii. $n=400, p=0.8$

Sneha Ravi
Sneha Ravi
Numerade Educator
01:22

Problem 7

For estimating a population characteristic, why is an unbiased statistic generally preferred over a biased statistic? Does unbiasedness alone guarantee that the estimate will be close to the actual value of the population characteristic? Explain.

Sara Sasani
Sara Sasani
Numerade Educator
01:43

Problem 8

Three different statistics are being considered for estimating a population characteristic. The sampling distributions of the three statistics are shown in the following illustration:
Which of these statistics are unbiased estimators of the population characteristic?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:21

Problem 9

Three different statistics are being considered for estimating a population characteristic. The sampling distributions of the three statistics are shown in the following illustration:
Which statistic would you recommend? Explain your choice.

Sara Sasani
Sara Sasani
Numerade Educator
01:23

Problem 10

A researcher wants to estimate the proportion of students enrolled at a university who are registered to vote. Would the standard error of the sample proportion $\hat{p}$ be larger if the actual population proportion was $p=0.4$ or $p=0.8 ?$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:08

Problem 11

Use the formula for the standard error of $\hat{p}$ to explain why increasing the sample size decreases the standard error.

Tyler Moulton
Tyler Moulton
Numerade Educator
02:28

Problem 12

A random sample will be selected from the population of all adult residents of a particular city. The sample proportion $\hat{p}$ will be used to estimate $p,$ the proportion of all adult residents who do not own a car. For which of the following situations will the estimate tend to be closest to the actual value of $p ?$
i. $n=500, p=0.1$
ii. $n=1000, p=0.2$
iii. $n=1200, p=0.3$

Sneha Ravi
Sneha Ravi
Numerade Educator
00:31

Problem 13

If two statistics are available for estimating a population characteristic, under what circumstances might you choose a biased statistic over an unbiased statistic?

Maxime Rossetti
Maxime Rossetti
Numerade Educator
02:23

Problem 14

A random sample will be selected from the population of all students enrolled at a large college. The sample proportion $\hat{p}$ will be used to estimate $p,$ the proportion of all students who use public transportation to travel to campus. For which of the following situations will the estimate tend to be closest to the actual value of $p ?$
i. $n=300, p=0.3$
ii. $n=700, p=0.2$
iii. $n=1000, p=0.1$

Kari Hasz
Kari Hasz
Numerade Educator
View

Problem 15

A researcher wants to estimate the proportion of city residents who favor spending city funds to promote tourism. Would the standard error of the sample proportion $\hat{p}$ be smaller for random samples of size $n=100$ or random samples of size $n=200 ?$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:16

Problem 16

A researcher wants to estimate the proportion of property owners who would pay their property taxes one month early if given a $\$ 50$ reduction in their tax bill. Would the standard error of the sample proportion $\hat{p}$ be larger if the actual population proportion were $p=0.2$ or if it were $p=0.4$ ?

Jameson Kuper
Jameson Kuper
Numerade Educator