Question 1

Use the following information for questions:
 In a past General Social Survey, 87% of a random sample of n = 990 respondents answered yes to the question "Would you approve of an adult male punching a stranger if the stranger had broken into the man's house?"
-A 90% confidence interval for the proportion of all Americans who approve of punching an intruder is
A) 0.852 to 0.888
B) 0.849 to 0.891
C) 0.845 to 0.895
D) 0.842 to 0.898

Accepted Answer

The correct confidence interval is calculated using the formula for a proportion's confidence interval, which is $p \pm z*\sqrt{\frac{p(1-p)}{n}}$, where $p$ is the sample proportion, $z$ is the z-score corresponding to the confidence level (in this case, 90% confidence level, which typically corresponds to a z-score of approximately 1.645), and $n$ is the sample size. Plugging in the given values ($p = 0.87$, $n = 990$), the calculation yields the interval 0.852 to 0.888, which matches option A.

Question 2

Use the following information for questions:
 In a past General Social Survey, 87% of a random sample of n = 990 respondents answered yes to the question "Would you approve of an adult male punching a stranger if the stranger had broken into the man's house?"
-A 98% confidence interval for the proportion of all Americans who approve of punching an intruder is
A) 0.852 to 0.888
B) 0.849 to 0.891
C) 0.845 to 0.895
D) 0.842 to 0.898

Accepted Answer

The confidence interval is calculated using the formula for a proportion confidence interval: p̂ ± Z*(√(p̂(1-p̂)/n)), where p̂ = 0.87, n = 990, and Z* for 98% confidence is approximately 2.33. This calculation results in the interval 0.845 to 0.895.

Question 3

Use the following information for questions:
 A 95% confidence interval for the proportion of young adults who skip breakfast is found to be 0.20 to 0.27.
-Which of the following is a correct interpretation of the 95% confidence level?
A) There is a 95% probability that the true proportion of young adults who skip breakfast is between 0.20 and 0.27.
B) In about 95% of all studies for which this procedure is used, the confidence interval will cover the true population proportion, but there is no way to know if this interval covers the true proportion or not.
C) If this study were to be repeated with a sample of the same size, there is a 95% probability that the sample proportion would be between 0.20 and 0.27.
D) The proportion of young adults who skip breakfast 95% of the time is between 0.20 and 0.27.

Accepted Answer

The correct interpretation of a 95% confidence interval is that in 95% of samples of the same size and from the same population, the calculated confidence intervals will contain the true population parameter. This does not mean there is a 95% probability that the true proportion lies within any given interval, as the true proportion is a fixed value and not a random variable.

Question 4

Use the following information for questions:
 A 95% confidence interval for the proportion of young adults who skip breakfast is found to be 0.20 to 0.27.
-Which of the following is the correct interpretation of the 95% confidence interval?
A) There is a 95% probability that the proportion of young adults who skip breakfast is between 0.20 and 0.27.
B) If this study were to be repeated with a sample of the same size, there is a 95% probability that the sample proportion would be between 0.20 and 0.27.
C) We can be 95% confident that the sample proportion of young adults who skip breakfast is between 0.20 and 0.27.
D) We can be 95% confident that the population proportion of young adults who skip breakfast is between 0.20 and 0.27.

Accepted Answer

The correct interpretation of a 95% confidence interval is that we are 95% confident that the true population parameter (in this case, the proportion of young adults who skip breakfast) falls within the interval (0.20, 0.27). This does not imply a probability statement about the parameter being in the interval in repeated samples, but rather a level of confidence in where the true parameter lies based on the sample data.

Question 5

Use the following information for questions:
 A 95% confidence interval for the proportion of young adults who skip breakfast is found to be 0.20 to 0.27.
-From the information provided, we can determine that
A) $\hat{p}$ = 0.235 and margin of error = 0.035.
B) $\hat{p}$ = 0.235 and margin of error = 0.07.
C) p = 0.235 and margin of error = 0.035.
D) p = 0.235 and margin of error = 0.07.

Accepted Answer

The sample proportion $\hat{p}$ is the midpoint of the confidence interval, which is $(0.20 + 0.27) / 2 = 0.235$. The margin of error is half the width of the confidence interval, which is $(0.27 - 0.20) / 2 = 0.035$.

Question 6

A poll is done to estimate the proportion of adult Americans who like their jobs. The poll is based on a random sample of 400 individuals. What is the "conservative" margin of error of this poll?
A) 0.10
B) 0.05
C) 0.04
D) 0.025

Accepted Answer

The answer of A poll is done to estimate the...

Question 7

A sample of n = 168 students was asked, "Do you believe in love at first sight?" The choices below show confidence intervals, in scrambled order, for 90%, 95%, 98%, and 99% confidence levels for the population proportion who would answer yes. Which choice gives the 99% confidence interval?
A) 0.56 to 0.68
B) 0.52 to 0.72
C) 0.53 to 0.71
D) 0.55 to 0.69

Accepted Answer

The 99% confidence interval is the widest interval because it reflects the highest level of confidence, meaning it accounts for the greatest amount of uncertainty. Among the given options, choice B (0.52 to 0.72) is the widest interval, indicating it is the 99% confidence interval.

Question 8

Suppose that 200 different polling organizations and academic researchers all do surveys in which the same question is asked. All 200 research groups construct a 95% confidence interval for the proportion who would say "yes"to this question. About how many of the 200 different 95% confidence intervals will not capture the true population proportion?
A) 0
B) 5
C) 10
D) 190

Accepted Answer

The answer of Suppose that 200 different polling organizations and...

Question 9

In a survey of n = 100 people, 10% are left-handed. Using the "exact"formula for margin of error, calculate the margin of error for a 95% confidence interval.
A) 0.03
B) 0.05
C) 0.06
D) 0.10

Accepted Answer

The answer of In a survey of n = 100...

Question 10

Suppose that 90%, 95%, 98%, and 99% confidence intervals are computed (using the same sample) for a population proportion. Which confidence level will give the narrowest interval?
A) 99%
B) 98%
C) 95%
D) 90%

Accepted Answer

The narrowest confidence interval is associated with the lowest confidence level because a lower confidence level means we are accepting a higher chance of error, thus requiring a smaller range to claim with that level of confidence. In this case, a 90% confidence interval will be narrower than 95%, 98%, or 99% confidence intervals.

Question 11

You plan to use a conservative margin of error of 2%. How large a sample size do you need?
A) 100
B) 400
C) 2500
D) None of the above

Accepted Answer

To achieve a conservative margin of error of 2% using the formula for margin of error in a proportion (which assumes a 50/50 split for maximum sample size), the sample size needed is $n = \left(\frac{1.96}{0.02}\right)^2 = 2401$, which is approximately 2500. This formula assumes a 95% confidence level.

Question 12

You plan to determine an approximate 95% confidence interval for a population proportion; you plan to use a conservative margin of error of 10%. How large a sample size do you need?
A) 100
B) 400
C) 2500
D) None of the above

Accepted Answer

The answer of You plan to determine an approximate 95%...

Question 13

Which of the following is the minimum sample size that could be used to guarantee that the margin of error for a confidence interval for a proportion is no more than 0.025?
A) 100
B) 400
C) 1600
D) 2500

Accepted Answer

The minimum sample size needed to guarantee a margin of error (E) no more than 0.025 for a confidence interval for a proportion can be calculated using the formula for the margin of error: $E = Z \sqrt{\frac{p(1-p)}{n}}$, where $Z$ is the Z-score corresponding to the desired confidence level, $p$ is the estimated proportion, and $n$ is the sample size. Without knowing the confidence level and assuming the worst-case scenario for $p$ (which maximizes the product $p(1-p)$) at $p = 0.5$, we simplify the formula to find $n$. For a maximum $E$ of 0.025 and using a Z-score for a common confidence level (e.g., 1.96 for 95% confidence), the calculation suggests a larger sample size is better for reducing the margin of error. Among the given options, 1600 is the minimum sample size that would typically be sufficient to ensure a margin of error no more than 0.025, even without specifying the confidence level, because the formula implies that a larger $n$ results in a smaller $E$, and 1600 is a common answer for such calculations with standard assumptions.

Question 14

You plan to determine an approximate 95% confidence interval for a population proportion; you plan to use a conservative margin of error of 5%. How large a sample size do you need?
A) 100
B) 400
C) 2500
D) None of the above

Accepted Answer

To determine a sample size for a 95% confidence interval with a margin of error of 5% (or 0.05) using a conservative estimate, we use the formula for the sample size of a proportion, which is $n = \left(\frac{Z^2 \cdot p \cdot (1-p)}{E^2}\right)$. For a conservative estimate, we use $p = 0.5$ because this maximizes the product $p(1-p)$, giving the largest sample size. With $Z = 1.96$ for a 95% confidence level and $E = 0.05$, the calculation is $n = \left(\frac{1.96^2 \cdot 0.5 \cdot 0.5}{0.05^2}\right) = 384.16$. Rounding up, we need at least 385 participants, which makes option B (400) the closest and correct choice.

Question 15

When creating a confidence interval for a proportion the conservative margin of error (studied in Chapter 5) and the margin of error found by taking 2 standard errors are closest to each other when
A) the true proportion p is close to 0.
B) the true proportion p is close to 1.
C) the true proportion p is close to .5.
D) the true proportion p is close to the sample proportion $\hat{p}$ .

Accepted Answer

The conservative margin of error and the margin of error found by taking 2 standard errors are closest to each other when the true proportion p is close to .5. This is because the maximum variability (and thus the maximum margin of error) occurs when p = 0.5 in a binomial distribution, which both methods aim to estimate.

Question 16

Suppose that 90% and 98% confidence intervals are both computed for a population proportion (using the same sample). Which confidence level will give the narrowest interval?
A) 90%
B) 98%
C) It depends on the value of the sample proportion.
D) It depends on the value of the true population proportion.

Accepted Answer

A 90% confidence interval will be narrower than a 98% confidence interval because a lower confidence level requires capturing less of the distribution, resulting in a smaller range. Higher confidence levels expand the interval to ensure a greater likelihood that the true parameter is captured.

Question 17

Use the following information for questions:
 A 95% confidence interval for p the proportion of supermarket customers who do not buy house-brand products at all is calculated and turns out to be: (0.18, 0.46).
-The point estimate of p equals 0.40.

Accepted Answer

The point estimate of p, which is the proportion of supermarket customers who do not buy house-brand products, should be the midpoint of the confidence interval, which is (0.18 + 0.46) / 2 = 0.32, not 0.40.

Question 18

Use the following information for questions:
 A 95% confidence interval for p the proportion of supermarket customers who do not buy house-brand products at all is calculated and turns out to be: (0.18, 0.46).
-The above confidence interval was constructed with a margin of error of 14 percentage points.

Accepted Answer

The margin of error is half the width of the confidence interval, so 14 percentage points is half of (0.46 - 0.18) = 0.28.

Question 19

Use the following information for questions:
 A 95% confidence interval for p the proportion of supermarket customers who do not buy house-brand products at all is calculated and turns out to be: (0.18, 0.46).
-The standard error of the sample proportion $\hat{p}$ is equal to 0.0714.

Accepted Answer

The standard error of the sample proportion is a component used in the calculation of confidence intervals, and the given value of 0.0714 could feasibly result in the provided confidence interval range of (0.18, 0.46) for a 95% confidence level, depending on the sample size and the specific formula used.

Question 20

Use the following information for questions:
 A 95% confidence interval for p the proportion of supermarket customers who do not buy house-brand products at all is calculated and turns out to be: (0.18, 0.46).
-There is a 95% probability that the population proportion of supermarket customers who do not buy house-brand products at all lies between 18% and 46%.

Accepted Answer

A 95% confidence interval means that if we were to take many samples and calculate a confidence interval for each, we would expect 95% of those intervals to contain the true population proportion. It does not mean there is a 95% probability that the true population proportion lies within any given interval.

Quiz 10: Estimating Proportions With Confidence

Use the following information for questions: A 95% confidence interval for the proportion of young adults who skip breakfast is found to be 0.20 to 0.27. -Which of the following is a correct interpretation of the 95% confidence level?

Use the following information for questions: A 95% confidence interval for the proportion of young adults who skip breakfast is found to be 0.20 to 0.27. -Which of the following is the correct interpretation of the 95% confidence interval?

Use the following information for questions: A 95% confidence interval for the proportion of young adults who skip breakfast is found to be 0.20 to 0.27. -From the information provided, we can determine that

A poll is done to estimate the proportion of adult Americans who like their jobs. The poll is based on a random sample of 400 individuals. What is the "conservative" margin of error of this poll?

A sample of n = 168 students was asked, "Do you believe in love at first sight?" The choices below show confidence intervals, in scrambled order, for 90%, 95%, 98%, and 99% confidence levels for the population proportion who would answer yes. Which choice gives the 99% confidence interval?

In a survey of n = 100 people, 10% are left-handed. Using the "exact"formula for margin of error, calculate the margin of error for a 95% confidence interval.

Suppose that 90%, 95%, 98%, and 99% confidence intervals are computed (using the same sample) for a population proportion. Which confidence level will give the narrowest interval?

You plan to use a conservative margin of error of 2%. How large a sample size do you need?

You plan to determine an approximate 95% confidence interval for a population proportion; you plan to use a conservative margin of error of 10%. How large a sample size do you need?

Which of the following is the minimum sample size that could be used to guarantee that the margin of error for a confidence interval for a proportion is no more than 0.025?

You plan to determine an approximate 95% confidence interval for a population proportion; you plan to use a conservative margin of error of 5%. How large a sample size do you need?

When creating a confidence interval for a proportion the conservative margin of error (studied in Chapter 5) and the margin of error found by taking 2 standard errors are closest to each other when

Suppose that 90% and 98% confidence intervals are both computed for a population proportion (using the same sample) . Which confidence level will give the narrowest interval?

Use the following information for questions: A 95% confidence interval for p the proportion of supermarket customers who do not buy house-brand products at all is calculated and turns out to be: (0.18, 0.46). -The point estimate of p equals 0.40.

Use the following information for questions: A 95% confidence interval for p the proportion of supermarket customers who do not buy house-brand products at all is calculated and turns out to be: (0.18, 0.46). -The standard error of the sample proportion $\hat{p}$ is equal to 0.0714.