AnswersNC-Math 4Sampling Distribution of the Sample Proportion

Sampling Distribution of the Sample Proportion Answers

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A professional tennis player has a serve-return rate of p = 0.71. A random sample of 55 serve returns is selected. Which of the following is the mean of the sampling distribution of ?

Question illustration
A
Mu Subscript p hat Baseline = p = 0.71
Option A
B
Mu Subscript p hat Baseline = n p = 55 (0.71) = 39.05
Option B
C
Mu Subscript p hat Baseline = 1 minus p = 1 minus 0.71 = 0.29
Option C
D
Mu Subscript p hat Baseline = n (1 minus p) = 55 (1 minus 0.71) = 15.95
Option D
2
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At a large, rural high school, 21% of sophomores have an allergy to ragweed, while 17% of seniors have one. Let and be the sample proportions of sophomores and seniors, respectively, who have an allergy to ragweed. Suppose 50 sophomores and 62 seniors from this school are selected at random and asked if they have an allergy to ragweed.Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of ?

Question illustration
A
The difference (sophomores – seniors) in the sample proportions of those who have an allergy to ragweed typically varies about 0.001 from the true difference in proportions.
B
The difference (sophomores – seniors) in the sample proportions of those who have an allergy to ragweed typically varies about 0.006 from the true difference in proportions.
C
The difference (sophomores – seniors) in the sample proportions of those who have an allergy to ragweed typically varies about 0.032 from the true difference in proportions.
D
The difference (sophomores – seniors) in the sample proportions of those who have an allergy to ragweed typically varies about 0.075 from the true difference in proportions.
3

Movie critics claim that 68% of adults and 79% of teenagers would recommend seeing the newest action movie. Suppose random samples of 43 adults and 52 teenagers are selected. Let and be the sample proportions of adult and teenage moviegoers, respectively, who would recommend this movie.Which of the following is the mean of the sampling distribution of ?

Question illustration
A
–0.11
B
0.11
C
0.74
D
1.47
4

A professional tennis player has a serve-return rate of p = 0.71. A random sample of 55 serve returns is selected. Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of ?

Question illustration
A
In SRSs of size 55, the sample proportion of this tennis player’s serve-return rate typically varies 0.004 from the true proportion p = 0.71.
Option A
B
In SRSs of size 55, the sample proportion of this tennis player’s serve-return rate typically varies 0.013 from the true proportion p = 0.71.
Option B
C
In SRSs of size 55, the sample proportion of this tennis player’s serve-return rate typically varies 0.061 from the true proportion p = 0.71.
Option C
D
In SRSs of size 55, the sample proportion of this tennis player’s serve-return rate typically varies 0.114 from the true proportion p = 0.71.
Option D
5

Movie critics claim that 68% of adults and 79% of teenagers would recommend seeing the newest action movie. Suppose random samples of 43 adults and 52 teenagers are selected. Let and be the sample proportions of adult and teenage moviegoers, respectively, who would recommend this movie.Which of the following is the correct shape and justification of the sampling distribution of ?

Question illustration
A
bimodal because one population proportion is centered at 0.68, while the other is centered at 0.79
B
approximately Normal because the expected numbers of successes and failures for each sample are all at least 10
C
not approximately Normal because the expected numbers of successes and failures for each sample are all at least 10
D
not approximately Normal because the expected numbers of successes and failures for the adult group are not both at least 10
6

In a nearby town, 56% of adults read nonfiction books as a hobby, while only 39% of teenagers read nonfiction books as a hobby. Let and be the sample proportions of adults and teenagers, respectively, who read nonfiction books as a hobby. Suppose 28 adults and 41 teenagers from this town are selected at random and asked if they read nonfiction books as a hobby.Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of ?

Question illustration
A
The difference (adult – teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about 0.003 from the true difference in proportions.
B
The difference (adult – teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about 0.015 from the true difference in proportions.
C
The difference (adult – teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about 0.055 from the true difference in proportions.
D
The difference (adult – teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about 0.121 from the true difference in proportions.
7

The proportion of twins born in a town is p = 0.12. Suppose we randomly select 100 women from this town who give birth in the next year. Which of the following is the mean of the sampling distribution of ?

Question illustration
A
Mu Subscript p hat = p = 0.12
Option A
B
Mu Subscript p hat = n p = 100 (0.12) = 12
Option B
C
Mu Subscript p hat = 1 minus p = 1 minus 0.12 = 0.88
Option C
D
Mu Subscript p hat = n (1 minus p) = 100 (1 minus 0.12) = 88
Option D
8

Agriculturists in a certain state claim that 43% of the residents in the northern portion of the state prefer flour tortillas over corn tortillas, while 59% of the residents in the southern portion of the state prefer flour tortillas over corn tortillas. Suppose random samples of 33 northerners and 41 southerners are selected. Let and be the sample proportions of northern and southern residents of this state, respectively, who would prefer flour tortillas over corn tortillas.Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of ?

Question illustration
A
The difference (northern – southern) in the sample proportions of those who prefer flour tortillas over corn tortillas typically varies about 0.013 from the true difference in proportions.
B
The difference (northern – southern) in the sample proportions of those who prefer flour tortillas over corn tortillas typically varies about 0.039 from the true difference in proportions.
C
The difference (northern – southern) in the sample proportions of those who prefer flour tortillas over corn tortillas typically varies about 0.115 from the true difference in proportions.
D
The difference (northern – southern) in the sample proportions of those who prefer flour tortillas over corn tortillas typically varies about 0.166 from the true difference in proportions.
9

Agriculturists in a certain state claim that 43% of the residents in the northern portion of the state prefer flour tortillas over corn tortillas, while 59% of the residents in the southern portion of the state prefer flour tortillas over corn tortillas. Suppose random samples of 33 northerners and 41 southerners are selected. Let and be the sample proportions of northern and southern residents of this state, respectively, who would prefer flour tortillas over corn tortillas.Which of the following is the mean of the sampling distribution of ?

Question illustration
A
–0.16
B
–0.08
C
0.16
D
0.51
10

Beet farmers claim that 76% of vegetarians like beets, while only 41% of carnivores like beets. Suppose a random samples of 21 vegetarians and 31 carnivores are asked if they like beets. Let and be the sample proportions of vegetarians and carnivores, respectively, who like beets.Which of the following is the correct shape and justification of the sampling distribution of ?

Question illustration
A
bimodal because one population proportion is centered at 0.76, while the other is centered at 0.41
B
approximately Normal because the expected numbers of successes and failures for each sample are all at least 10
C
not approximately Normal because the expected numbers of successes and failures for each sample are all at least 10
D
not approximately Normal because the expected numbers of successes and failures for the vegetarian group are not both at least 10

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