Deck 7: Estimates and Sample Sizes

Question 1

Provide an appropriate response.
Draw a diagram of the chi-square distribution. Discuss its shape and values.

Answer 1

The chi-square distribution is non-symmetric and skewed to the right. The values are 0 and positive.

Question 2

Provide an appropriate response.

-When determining the sample size for a desired margin of error, the formula is $\mathrm{n}=\frac{\left[{ }^{\mathrm{z}} \alpha / 2\right]^{2} \cdot \mathrm{\hat{p}\hat{q}}}{\mathrm{E}^{2}}$ . Based on this formula, discuss the fact that sample size is not dependent on the population size; that is, it is not necessary to sample a particular percent of the population.

Answer 2

As shown in the formula, the appropriate sample size is dependent on the appropriate z score, the sample proportion, and the margin of error, not on N, the population size.

Question 3

Provide an appropriate response.

-When determining sample size we need to know $\hat { p }$ If we have no prior information, what are two methods that can be used?

Answer 3

$$\text { Use a result from a prior study or use } \hat{p} = 0.5 \text {. }$$

Question 4

Complete the table to compare z and t distributions. $$\begin{array} { | l | l | l | } \hline & \text { z distribution } & \text { t distribution } \\\hline \text { Shape } & & \\\hline \text { Mean value } & & \\\hline \text { Standard deviation value } & & \\\hline \text { Requirements } & & \\\hline\end{array}$$

Question 5

Provide an appropriate response.
Define confidence interval and degree of confidence. Make up an example of a confidence interval and interpret the result.

Question 6

Provide an appropriate response.

Question 7

Provide an appropriate response.
Explain how confidence intervals might be used to make decisions. Give an example to clarify your explanation.

Question 8

Why would manufacturers and businesses be interested in constructing a confidence interval for the population variance? Would manufacturers and businesses want large or small variances?

Question 9

Provide an appropriate response.

Question 10

Provide an appropriate response.

Question 11

Provide an appropriate response.
Describe the steps for finding a confidence interval.

Question 12

Provide an appropriate response.
Interpret the following 95% confidence interval for mean weekly salaries of shift managers at Guiseppe's Pizza and Pasta.

Question 13

Provide an appropriate response.
Describe the process for finding the confidence interval for a population proportion.

Question 14

Provide an appropriate response.
What is the best point estimate for the population proportion? Explain why that point estimate is best.

Question 15

The Bide-a-While efficiency hotel, which caters to business workers who stay for extended periods of time (weeks or months), offers room service. In a small study of 35 randomly selected room service orders, the 95% confidence interval for mean delivery time for room service is 24.8 < µ < 29.6 minutes. The marketing director is trying to determine if she can advertise "room service in under 30 minutes, or the order is free." How would you advise her?

Question 16

Provide an appropriate response.
What assumption about the parent population is needed to use the t distribution to compute the margin of error?

Question 17

Provide an appropriate response.
Define a point estimate. What is the best point estimate for

Question 18

Provide an appropriate response.

Question 19

Provide an appropriate response.
Explain the difference between descriptive and inferential statistics.

Question 20

Provide an appropriate response.
Identify the correct distribution (z, t, or neither)for each of the following.

Question 21

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.011$ ; confidence level: $92 \% ; \hat { \mathrm { p } }$ and $\hat { \mathrm { q } }$ unknown

A) 40
B) 1
C) 6328
D) 6327

Question 22

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p

-n = 133, x = 82; 90 percent

A)0.548 < p < 0.686
B)0.546 < p < 0.688
C)0.551 < p < 0.683
D)0.550 < p < 0.684

Question 23

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p

-n = 102, x = 52; 88 percent

A)0.432 < p < 0.588
B)0.428 < p < 0.592
C)0.433 < p < 0.587
D)0.429 < p < 0.591

Question 24

$$\text { Find the critical value } \mathrm { z } _ { \alpha / 2 } \text { that corresponds to a degree of confidence of } 98 \% \text {. }$$

A)2.05
B)1.75
C)2.33
D)2.575

Question 25

The following confidence interval is obtained for a population proportion, p: 0.855 < p < 0.897 Use these confidence interval limits to find the margin of error, E.

A)0.021
B)0.876
C)0.022
D)0.042

Question 26

Find the margin of error for the 95% confidence interval used to estimate the population proportion

-In a clinical test with 2353 subjects, 1136 showed improvement from the treatment.

A)0.0273
B)0.0172
C)0.0226
D)0.0202

Question 27

The following confidence interval is obtained for a population proportion, p: (0.348, 0.382) Use these confidence interval limits to find the margin of error, E.

A)0.017
B)0.034
C)0.015
D)0.018

Question 28

The following confidence interval is obtained for a population proportion, p: (0.298, 0.338) Use these confidence interval limits to find the point estimate, $\hat{\mathrm { p }}$ .

A) $0.318$
B) $0.321$
C) $0.298$
D) $0.338$

Question 29

Express the confidence interval in the form of $\hat{p}$ ± E.

-0.02 < p < 0.48

A) $\hat { p } = 0.23 \pm 0.5$
B) $\hat { p } = 0.25 - 0.23$
C) $\hat { p } = 0.25 \pm 0.23$
D) $\hat { p } = 0.25 \pm 0.5$

Question 30

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.002$ ; confidence level: $93 \% ; \hat { \mathrm { p } }$ and $\hat{\mathrm { q }}$ unknown

A) 410
B) 204,757
C) 409
D) 204,756

Question 31

$$\text { Find the value of } - \mathrm { z } _{\alpha / 2} \text { that corresponds to a level of confidence of } 98.22 \text { percent. }$$

A)-2.1
B)2.37
C)0.0089
D)-2.37

Question 32

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p

-n = 182, x = 135; 95 percent

A)0.690 < p < 0.793
B)0.691 < p < 0.792
C)0.677 < p < 0.806
D)0.678 < p < 0.805

Question 33

The following confidence interval is obtained for a population proportion, p: 0.802 < p < 0.828 Use these confidence interval limits to find the point estimate, $\hat { p }$ .

A) $0.818$
B) $0.812$
C) $0.802$
D) $0.815$

Question 34

Find the margin of error for the 95% confidence interval used to estimate the population proportion

-n = 169, x = 107

A)0.127
B)0.0654
C)0.00269
D)0.0727

Question 35

Find the margin of error for the 95% confidence interval used to estimate the population proportion

-n = 230, x = 90

A)0.0663
B)0.0631
C)0.0757
D)0.0568

Question 36

Find the margin of error for the 95% confidence interval used to estimate the population proportion

-In a survey of 4100 T.V. viewers, 20% said they watch network news programs.

A)0.0122
B)0.00915
C)0.0140
D)0.0160

Question 37

Express the confidence interval in the form of $\hat{p}$ ± E.

- $- 0.128 < p < 1.212$

A) $\hat { p } = 0.542 \pm 0.4$
B) $\hat { p } = 0.542 - 0.67$
C) $\hat { p } = 0.67 \pm 0.4$
D) $\hat { p } = 0.542 \pm 0.67$

Question 38

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p

-n = 84, x = 37; 98 percent

A)0.313 < p < 0.567
B)0.333 < p < 0.547
C)0.334 < p < 0.546
D)0.314 < p < 0.566

Question 39

$$\text { Find the critical value } \mathrm { z } \alpha / 2 \text { that corresponds to a degree of confidence of } 91 \% \text {. }$$

A)1.75
B)1.34
C)1.645
D)1.70

Question 40

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p

-n = 58, x = 28; 95 percent

A)0.353 < p < 0.613
B)0.375 < p < 0.591
C)0.354 < p < 0.612
D)0.374 < p < 0.592

Question 41

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-Of 346 items tested, 12 are found to be defective. Construct the 98% confidence interval for the proportion of all such items that are defective.

A)0.0154 < p < 0.0540
B)0.0118 < p < 0.0576
C)0.0345 < p < 0.0349
D)0.0110 < p < 0.0584

Question 42

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.05$ ; confidence level: $95 \% ; \hat { \mathrm { p } }$ and $\hat { q }$ unknown

A) 197
B) 10
C) 384
D) 385

Question 43

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.045$ ; confidence level: $96 \% ; \hat { p }$ and $\hat { q }$ are unknown

A) 519
B) 24
C) 12
D) 518

Question 44

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-A survey of 865 voters in one state reveals that 408 favor approval of an issue before the legislature. Construct the 95% confidence interval for the true proportion of all voters in the state who favor approval.

A)0.438 < p < 0.505
B)0.435 < p < 0.508
C)0.444 < p < 0.500
D)0.471 < p < 0.472

Question 45

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-When 306 college students are randomly selected and surveyed, it is found that 115 own a car. Find a 99% confidence interval for the true proportion of all college students who own a car.

A)0.305 < p < 0.447
B)0.311 < p < 0.440
C)0.330 < p < 0.421
D)0.322 < p < 0.430

Question 46

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: 0.04; confidence level: 99%; from a prior study, ^p is estimated by 0.08

A)306
B)12
C)177
D)367

Question 47

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.01$ ; confidence level: $95 \%$ ; from a prior study, $\hat { \mathrm { p } }$ is estimated by the decimal equivalent of $69 \%$

A)8218
B)14,184
C)26,507
D)7396

Question 48

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-Of 86 adults selected randomly from one town, 63 have health insurance. Find a 90% confidence interval for the true proportion of all adults in the town who have health insurance.

A)0.621 < p < 0.844
B)0.654 < p < 0.811
C)0.639 < p < 0.826
D)0.610 < p < 0.855

Question 49

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.04$ ; confidence level: $95 \%$ ; from a prior study, $\hat { \mathrm { p } }$ is estimated by the decimal equivalent of $92 \%$ ,

A)177
B)531
C)7
D)157

Question 50

Solve the problem.
Find the point estimate of the true proportion of people who wear hearing aids if, in a random sample of 875 people, 56 people had hearing aids.

A)0.063
B)0.060
C)0.064
D)0.936

Question 51

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.006$ ; confidence level: $96 \% ; \hat { p }$ and $\mathrm { q }$ are unknown

A) 176
B) 29,184
C) 86
D) 29,185

Question 52

Solve the problem.
464 randomly selected light bulbs were tested in a laboratory, 424 lasted more than 500 hours. Find a point estimate of the true proportion of all light bulbs that last more than 500 hours.

A)0.914
B)0.912
C)0.477
D)0.086

Question 53

Solve the problem.
30 randomly picked people were asked if they rented or owned their own home, 30 said they rented. Obtain a point estimate of the true proportion of home owners.

A)0.500
B)0.000
C)1.000
D)0.033

Question 54

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.007$ ; confidence level: $99 \%$ ; from a prior study, $\hat { \mathrm { p } }$ is estimated by $0.238$

A) 24,541
B) 14,218
C) 172
D) 22,087

Question 55

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.07$ ; confidence level: $90 \%$ ; from a prior study, $\hat{\mathrm { p }}$ is estimated by $0.19$ .

A) 255
B) 6
C) 85
D) 75

Question 56

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.025$ ; confidence level: $96 \% ; \hat { \mathrm { p } }$ and $\hat { \mathrm { q } }$ are unknown

A) 1680
B) 21
C) 43
D) 1681

Question 57

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-A survey of 300 union members in New York State reveals that 112 favor the Republican candidate for governor. Construct the 98% confidence interval for the true population proportion of all New York State union members who favor the Republican candidate.

A)0.304 < p < 0.442
B)0.308 < p < 0.438
C)0.316 < p < 0.430
D)0.301 < p < 0.445

Question 58

Solve the problem.

-50 people are selected randomly from a certain population and it is found that 20 people in the sample are over 6 feet tall. What is the point estimate of the true proportion of people in the population who are over 6 feet tall?

A)0.40
B)0.24
C)0.60
D)0.50

Question 59

Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error around the population p

-Margin of error: $0.005$ ; confidence level: $97 \%$ ; $\mathrm { p }$ and $\mathrm { q }$ are unknown

A) 753,425
B) 47,088
C) 47,089
D) 188,357

Question 60

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-Of 357 randomly selected medical students, 30 said that they planned to work in a rural community. Find a 95% confidence interval for the true proportion of all medical students who plan to work in a rural community.

A)0.0462 < p < 0.122
B)0.0599 < p < 0.108
C)0.0553 < p < 0.113
D)0.0498 < p < 0.118

Question 61

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.

-The sample size is $n = 11 , \sigma$ is not known, and the original population is normally distributed.

A) No
B) Yes

Question 62

Use the confidence level and sample data to find the margin of error E

-College students' annual earnings: $99 \%$ confidence; $n = 71 , \bar { x } = \$ 3660 , \sigma = \$ 879$

A) $\$ 1118$
B) $\$ 243$
C) $\$ 269$
D) $\$ 8$

Question 63

Solve the problem.

-Suppose we wish to construct a confidence interval for a population proportion p. If we sample without replacement from a relatively small population of size N, the margin of error E is modified to include the finite population correction factor as follows: $$E = z _ { \alpha } / 2 \sqrt { \frac { \hat { p }\hat { q } } { n } } \sqrt { \frac { N - n } { N - 1 } }$$ Construct a 90% confidence interval for the proportion of students at a school who are left handed. The number of students at the school is N = 400. In a random sample of 82 students, selected without replacement, there are 11 left handers.

A)0.091 < p < 0.177
B)0.086 < p < 0.182
C)0.072 < p < 0.196
D)0.079 < p < 0.189

Question 64

Solve the problem.

-A researcher is interested in estimating the proportion of voters who favor a tax on e-commerce. Based on a sample of 250 people, she obtains the following 99% confidence interval for the population proportion p:
0.113 < p < 0.171
Which of the statements below is a valid interpretation of this confidence interval?
A: There is a 99% chance that the true value of p lies between 0.113 and 0.171.
B: If many different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, 99% of the time the true value of p would lie between 0.113 and 0.171.
C: If many different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, in the long run 99% of the confidence intervals would contain the true value of p.
D: If 100 different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, exactly 99 of these confidence intervals would contain the true value of p.

A)A
B)D
C)B
D)C

Question 65

Use the confidence level and sample data to find the margin of error E

-Weights of eggs: $95 \%$ confidence; $\mathrm { n } = 47 , \overline { \mathrm { x } } = 1.44 \mathrm { oz } , \sigma = 0.39 \mathrm { oz }$

A) $6.86 \mathrm { oz }$
B) $0.09 \mathrm { oz }$
C) $0.02 \mathrm { oz }$
D) $0.11 \mathrm { oz }$

Question 66

Solve the problem.

-Find the critical valu $\mathrm { z } _ { \alpha / 2 }$ that corresponds to a degree of confidence of 98%.

A)2.33
B)2.05
C)1.75
D)2.575

Question 67

Solve the problem.
In a certain population, body weights are normally distributed with a mean of 152 pounds and a standard deviation of 26 pounds. How many people must be surveyed if we want to estimate the percentage who weigh more than 180 pounds? Assume that we want 96% confidence that the error is no more than 2 percentage points.

A)923
B)1267
C)2001
D)2628

Question 68

Solve the problem.

-A one-sided confidence interval for $\mathrm { p }$ can be written as $\mathrm { p } < \hat { \mathrm { p } } + \mathrm { E }$ or $\mathrm { p } > \hat { \mathrm { p } } - \mathrm { E }$ where the margin of error $\mathrm { E }$ is modified by replacing $\mathrm { z } _ { \alpha / 2 }$ with $\mathrm { z } _ { \alpha }$ . If a teacher wants to report that the fail rate on a test is at most $x$ with $90 \%$ confidence, construct the appropriate one-sided confidence interval. Assume that a simple random sample of 58 students results in 5 who fail the test.

A) $p > 0.039$
B) $p < 0.133$
C) $\mathrm { p } < 0.147$
D) $p < 0.039$

Question 69

Use the confidence level and sample data to find the margin of error E

-Replacement times for washing machines: $90 \%$ confidence; $\mathrm { n } = 36 , \bar{\mathrm { x} } = 10.0$ years, $\sigma = 2.1$ years

A) $0.4$ years
B) $6.0$ years
C) $0.1$ years
D) $0.6$ years

Question 70

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-Of 129 randomly selected adults, 32 were found to have high blood pressure. Construct a 95% confidence interval for the true percentage of all adults that have high blood pressure.

A)15.9% < p < 33.7%
B)17.4% < p < 32.3%
C)18.5% < p < 31.1%
D)15.0% < p < 34.6%

Question 71

Determine whether the given conditions justify using the margin of error $\mathrm { E } = \mathrm { z } _ { \alpha / 2 } \sigma / \sqrt { \mathrm { n } }$ when finding a confidence interval estimate of the population mean $\mu$ .

-The sample size is $\mathrm { n } = 10$ and $\sigma$ is not known.

A) No
B) Yes

Question 72

Solve the problem.
A newspaper article about the results of a poll states: "In theory, the results of such a poll, in 99 cases out of 100 should differ by no more than 6 percentage points in either direction from what would have been obtained by interviewing all voters in the United States." Find the sample size suggested by this statement.

A)268
B)19
C)461
D)378

Question 73

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-Of 243 employees selected randomly from one company, 14.81% of them commute by carpooling. Construct a 90% confidence interval for the true percentage of all employees of the company who carpool.

A)9.50% < p < 20.1%
B)10.3% < p < 19.3%
C)8.94% < p < 20.7%
D)11.1% < p < 18.6%

Question 74

Solve the problem.

-Find the critical value $\mathrm { z } _ { \alpha / 2 }$ that corresponds to a degree of confidence of 91%.

A)1.70
B)1.75
C)1.34
D)1.645

Question 75

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-A study involves 634 randomly selected deaths, with 32 of them caused by accidents. Construct a 98% confidence interval for the true percentage of all deaths that are caused by accidents.

A)3.61% < p < 6.48%
B)3.02% < p < 7.07%
C)2.80% < p < 7.29%
D)3.34% < p < 6.75%

Question 76

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.


-Of 132 adults selected randomly from one town, 33 of them smoke. Construct a 99% confidence interval for the true percentage of all adults in the town that smoke.

A)16.2% < p < 33.8%
B)18.8% < p < 31.2%
C)15.3% < p < 34.7%
D)17.6% < p < 32.4%

Question 77

Solve the problem.

-Find the value of ${ } ^ { Z } \alpha / 2$ that corresponds to a level of confidence of 98.22 percent.

A)2.1
B)2.37
C)0.0089
D)-2.37

Question 78

Solve the problem.

-Suppose that n trials of a binomial experiment result in no successes. According to the "Rule of Three", we have 95% confidence that the true population proportion has an upper bound of 3/n. If a manufacturer randomly selects 21 computers for quality control and finds no defective computers, what statement can you make by using the rule of three, about the proportion p, of all its computers which are defective?

A) We are $95 \%$ confident that $p$ is greater than $\frac { 1 } { 7 }$ .
B) We are $95 \%$ confident that $p$ does not exceed $\frac { 1 } { 7 }$ .
C) We are $95 \%$ confident that $p$ lies between $\frac { 1 } { 21 }$ and $\frac { 1 } { 7 }$ .
D) The value of $p$ cannot be greater than $\frac { 1 } { 7 }$ .

Question 79

Determine whether the given conditions justify using the margin of error $\mathrm { E } = \mathrm { z } _ { \alpha / 2 } \sigma / \sqrt { \mathrm { n } }$ when finding a confidence interval estimate of the population mean $\mu$ .

-The sample size is n = 286 and $\sigma = 15$

A)Yes
B)No

Question 80

Determine whether the given conditions justify using the margin of error $\mathrm { E } = \mathrm { z } _ { \alpha / 2 } \sigma / \sqrt { \mathrm { n } }$ when finding a confidence interval estimate of the population mean $\mu$ .

-The sample size is n $\mathrm { n } = 5 , \sigma = 12.7$ and the original population is normally distributed.

A)Yes
B)No