Question 1

For the given combination of $\alpha$ and degrees of freedom (df), find the value of $\chi^2 _ { ( 1 - \alpha / 2 ) }$ that would be used to find the upper endpoint of a confidence interval for $\sigma ^ { 2 }$.
$\alpha = 0.05 \text {, df } = 25$
A) $13.1197$
B) $14.6114$
C) $12.4011$
D) $40.6465$

Accepted Answer

The answer of For the given combination of $\alpha$ and...

Question 2

If no estimate of p exists when determining the sample size for a confidence interval for a proportion, we can use .5 in the formula to get a value for n.

Accepted Answer

This is correct. When no estimate for p is available, the value of p is assumed to be 0.5 (closest to 0.5) in order to get the largest possible sample size. This ensures that the resulting confidence interval will have the desired level of precision.

Question 3

It is desired to estimate the proportion of college students who feel a sudden relief now that their statistics class is over. How many students must be sampled in order to estimate the true proportion to within 2% at the 90% confidence level?
A) n = 189
B) n = 2401
C) n = 1692
D) n = 133
E) Cannot determine because no estimate of p or q exists in this problem

Accepted Answer

The answer of It is desired to estimate the proportion...

Question 4

A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. If the dean wanted to estimate the proportion of all students receiving financial aid to within 3% with 90% reliability, how many students would need to be sampled?
A) 728
B) 176
C) 443
D) 22

Accepted Answer

To estimate the proportion with a margin of error of 3% and 90% confidence, the sample size can be calculated using the formula for sample size in proportion estimation: n = (Z^2 * p * (1-p)) / E^2. Here, Z is the Z-score for 90% confidence (approximately 1.645), p is the sample proportion (118/200 = 0.59), and E is the margin of error (0.03). Plugging in these values gives a sample size of approximately 728.

Question 5

For the given combination of $\alpha$ and degrees of freedom (df), find the value of $\chi { } _ { \alpha / 2 } ^ { 2 }$ that would be used to find the lower endpoint of a confidence interval for $\sigma ^ { 2 }$.
$$\alpha = 0.01 , \mathrm { df } = 30$$
A) $53.6720$
B) $50.8922$
C) $52.3356$
D) $13.7867$

Accepted Answer

The answer of For the given combination of $\alpha$ and...

Question 6

Given the values of $\bar { x } , \mathrm {~s}$, and $\mathrm { n }$, form a $99 \%$ confidence interval for $\sigma$. $\overline { \mathrm { x } } = 11.8 , \mathrm {~s} = 9.8 , \mathrm { n } = 8$
A) $( 5.76,26.07 )$
B) $( 6.03,23.29 )$
C) $( 33.15,679.58 )$
D) $( 3.38,69.34 )$

Accepted Answer

The answer of Given the values of \(\bar { x...

Question 7

A previous random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. How many citizens would need to be sampled for a 99% confidence interval to estimate the true proportion within 5%?
A) 653
B) 690
C) 664
D) 611

Accepted Answer

The answer of A previous random sample of 4000 U.S....

Question 8

Given the values of $\bar { x } , s$, and $n$, form a $99 \%$ confidence interval for $\sigma ^ { 2 }$.
$$\overline { \mathrm { x } } = 10.2 , \mathrm {~s} = 8.6 , \mathrm { n } = 8$$
A) $( 25.53,523.34 )$
B) $( 28.02,417.84 )$
C) $( 29.18,598.1 )$
D) $( 3.26,48.59 )$

Accepted Answer

The answer of Given the values of \(\bar { x...

Question 9

Suppose you wanted to estimate a binomial proportion, p, correct to within .05 with probability 0.98. What size sample would need to be selected if p is known to be approximately 0.7?

Accepted Answer

The answer of Suppose you wanted to estimate a binomial...

Question 10

Suppose it is desired to estimate the average time a customer spends in a particular store to within 5 minutes (e.g., + 5 minutes) at 97% reliability. It is estimated that the standard deviation of the times is 15 minutes. How large a sample should be taken to get the desired interval?
A) n = 6.51
B) n = 150
C) n = 43
D) n = 7

Accepted Answer

To find the sample size, use the formula n = (Z*σ/E)^2, where Z is the z-score for 97% confidence (approximately 2.17), σ is the standard deviation (15 minutes), and E is the margin of error (5 minutes). Calculating gives n ≈ 43.

Question 11

One way of reducing the width of a confidence interval is to reduce the size of the sample taken.

Accepted Answer

Reducing the size of the sample typically increases the width of the confidence interval because a smaller sample size increases the uncertainty (or standard error) of the estimate.

Question 12

Sales of a new line of athletic footwear are crucial to the success of a company. The company wishes to estimate the average weekly sales of the new footwear to within $500 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1200. How many weeks of data must be sampled for the company to get the information it desires?
A) 16 weeks
B) 4 weeks
C) 10 weeks
D) 7794 weeks

Accepted Answer

To find the sample size (n), use the formula for the sample size of a mean: n = (Z*σ/E)^2, where Z is the Z-score for 90% confidence (1.645), σ is the standard deviation ($1200), and E is the margin of error ($500). Plugging in the values: n = (1.645*1200/500)^2 ≈ 15.6, which rounds up to 16 weeks.

Question 13

The standard deviation of a population is estimated to be 345 units. To estimate the population mean to within 50 units with 98% reliability, what size sample should be selected?

Accepted Answer

The answer of The standard deviation of a population is...

Question 14

The director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, $\bar { x } = 19.8 \text { and } s ^ { 2 } = 9$ . If the director wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 95% reliability, what is the minimum sample size she should use?
A) 35
B) 312
C) 18
D) 159

Accepted Answer

The answer of The director of a hospital wishes to...

Question 15

After elections were held, it was desired to estimate the proportion of voters who regretted that they did not vote. How many voters must be sampled in order to estimate the true proportion to within 2% (e.g., + 0.02) at the 90% confidence level? Assume that we believe this proportion lies close to 30%.
A) n = 2017
B) n = 2401
C) n = 1692
D) n = 1421
E) Cannot determine because no estimate of p or q exists in this problem.

Accepted Answer

The answer of After elections were held, it was desired...

Question 16

Sales of a new line of athletic footwear are crucial to the success of a newly formed company. The company wishes to estimate the average weekly sales of the new footwear to within $150 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1650. How many weeks of data must be sampled for the company to get the information it desires?

Accepted Answer

The answer of Sales of a new line of athletic...

Question 17

A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers. How large a sample do the buyers need in order to be 90% confident that the margin of error of their estimate is within 5%?

Accepted Answer

The answer of A local men's clothing store is being...

Question 18

In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a __________ interval.
A) narrower
B) wider
C) less significant
D) biased

Accepted Answer

An increase in sample size will lead to a narrower interval as it reduces the margin of error and increases the precision of the estimate.

Question 19

It is desired to estimate the average time it takes Statistics students to finish a computer project to within two hours at 97% reliability. It is estimated that the standard deviation of the times is 14 hours. How large a sample should be taken to get the desired interval?
A) n = 17
B) n = 189
C) n = 231
D) n = 133

Accepted Answer

To find the sample size, use the formula for the sample size of a mean: n = (Z * σ / E)^2. Here, Z is the z-value for 97% confidence (approximately 2.17), σ is the standard deviation (14 hours), and E is the margin of error (2 hours). Plugging these values into the formula gives n ≈ 231.

Question 20

A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following confidence interval: (.438, .642). Using the information above, what sample size would be necessary if we wanted to estimate the true proportion to within 1% using 95% reliability?
A) 9543
B) 9604
C) 9989
D) 9220

Accepted Answer

The answer of A confidence interval was used to estimate...

Quiz 7: Inferences Based on a Single Sample: Estimation With Confidence Intervals

For the given combination of $\alpha$ and degrees of freedom (df), find the value of $\chi^2 _ { ( 1 - \alpha / 2 ) }$ that would be used to find the upper endpoint of a confidence interval for $\sigma ^ { 2 }$ . $\alpha = 0.05 \text {, df } = 25$ A) $13.1197$ B) $14.6114$ C) $12.4011$ D) $40.6465$

If no estimate of p exists when determining the sample size for a confidence interval for a proportion, we can use .5 in the formula to get a value for n.

It is desired to estimate the proportion of college students who feel a sudden relief now that their statistics class is over. How many students must be sampled in order to estimate the true proportion to within 2% at the 90% confidence level?

Given the values of $\bar { x } , \mathrm {~s}$ , and $\mathrm { n }$ , form a $99 \%$ confidence interval for $\sigma$ . $\overline { \mathrm { x } } = 11.8 , \mathrm {~s} = 9.8 , \mathrm { n } = 8$ A) $( 5.76,26.07 )$ B) $( 6.03,23.29 )$ C) $( 33.15,679.58 )$ D) $( 3.38,69.34 )$

A previous random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. How many citizens would need to be sampled for a 99% confidence interval to estimate the true proportion within 5%?

Given the values of $\bar { x } , s$ , and $n$ , form a $99 \%$ confidence interval for $\sigma ^ { 2 }$ . $\overline { \mathrm { x } } = 10.2 , \mathrm {~s} = 8.6 , \mathrm { n } = 8$ A) $( 25.53,523.34 )$ B) $( 28.02,417.84 )$ C) $( 29.18,598.1 )$ D) $( 3.26,48.59 )$

Suppose you wanted to estimate a binomial proportion, p, correct to within .05 with probability 0.98. What size sample would need to be selected if p is known to be approximately 0.7?

Suppose it is desired to estimate the average time a customer spends in a particular store to within 5 minutes (e.g., + 5 minutes) at 97% reliability. It is estimated that the standard deviation of the times is 15 minutes. How large a sample should be taken to get the desired interval?

One way of reducing the width of a confidence interval is to reduce the size of the sample taken.

The standard deviation of a population is estimated to be 345 units. To estimate the population mean to within 50 units with 98% reliability, what size sample should be selected?

In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a __________ interval.

It is desired to estimate the average time it takes Statistics students to finish a computer project to within two hours at 97% reliability. It is estimated that the standard deviation of the times is 14 hours. How large a sample should be taken to get the desired interval?