Question 1

Statisticians like precision in their interval estimates.A low margin of error is needed to achieve this.Which of the following supports this when selecting sample sizes?
A)A larger sample size reduces the margin of error.
B)A smaller sample size reduces the margin of error.
C)A larger sample size increases the margin of error.
D)A sample size has no impact on the margin of error.

Accepted Answer

A larger sample size reduces the margin of error because it allows for a more representative sample of the population, reducing the likelihood of sampling error.

Question 2

Exhibit 8.4.According to a report in USA Today (February 1,2012),more and more parents are helping their young adult children get homes.Suppose 8 persons in a random sample of 40 young adults who recently purchased a home in Kentucky got help from their parents.You have been asked to construct a 95% confidence interval for the population proportion of all young adults in Kentucky who got help from their parents. Refer to Exhibit 8.4.What is the margin of error for a 95% confidence interval for the population proportion?
A)1.645(0.0040)
B)1.645(0.0632)
C)1.96(0.0040)
D)1.96(0.0632)

Accepted Answer

The margin of error for a 95% confidence interval for a population proportion is calculated using the formula: z * sqrt(p(1-p)/n). Here, z is 1.96 for a 95% confidence level, p is the sample proportion (8/40 = 0.2), and n is the sample size (40). The correct choice uses 1.96 and the correct standard error calculation.

Question 3

Find the minimum sample size when we want to construct a 90% confidence interval on the population proportion for the support of candidate A in the following mayoral election.Candidate A is facing two opposing candidates.In a preselected poll of 100 residents,22 supported her.The desired margin of error is 0.08.
A)n = 44
B)n = 72
C)n = 73
D)n = 103

Accepted Answer

We can use the formula for the margin of error of a population proportion:
ME = z*sqrt(p*(1-p)/n)
where ME = 0.08, z is the z-score for a 90% confidence interval (1.645), and p hat is the sample proportion (22/100 = 0.22).
Solving for n, we get:
n = ((z^2)*p*(1-p))/(ME^2)
n = ((1.645^2)*(0.22)*(1-0.22))/(0.08)^2
n ≈ 72.56
We round up to the nearest whole number to ensure that the margin of error is at most 0.08, so the minimum sample size is n = 73. Therefore, the correct choice is C.

Question 4

Exhibit 8-6.A sample of 1400 American households was asked if they planned to buy a new car next year.Of the respondents,34% indicated they planned to buy a new car next year. Refer to Exhibit 8-6.Construct a 98% confidence interval of the proportion of American households who expect to buy a new car next year.
A)[0.3074,0.3726]
B)[0.3105,0.3695]
C)[0.3140,0.3660]
D)[0.3392,0.3408]

Accepted Answer

To calculate the confidence interval, we can use the formula:

CI = p̂ ± zα/2 √((p̂*(1-p̂))/n)

where: 
p̂ = sample proportion = 0.34
n = sample size = 1400
zα/2 = critical value at 98% confidence level = 2.33

Plugging in the values, we get:

CI = 0.34 ± 2.33 √((0.34*(1-0.34))/1400)
CI = 0.34 ± 0.0295
CI = (0.3105, 0.3695)

Therefore, the answer is B, the 98% confidence interval of the proportion of American households who expect to buy a new car next year is (0.3105, 0.3695).

Question 5

Exhibit 8-5.A sample of 2,007 American adults was asked how they viewed China,with 17% of respondents indicating "unfriendly" and 6% of respondents indicating "an enemy" (Gallup,Nov.30 - Dec.18,2011). Refer to Exhibit 8-5.Construct a 95% confidence interval of the proportion of American adults who view China as either "unfriendly" or as "an enemy."
A)[0.2013,0.2587]
B)[0.2116,0.2484]
C)[0.2146,0.2454]
D)[0.2208,0.2392]

Accepted Answer

The combined proportion of respondents who view China as either "unfriendly" or "an enemy" is 23% (0.23). Using this proportion, the sample size (n=2007), and the standard formula for a confidence interval for a proportion, the 95% confidence interval is calculated as [0.2116, 0.2484].

Question 6

Construct a 95% confidence interval on the population proportion for the support of candidate A in the following mayoral election.Candidate A is facing two opposing candidates in a mayoral election.In a recent poll of 300 residents,she has garnered 51% support.
A)[0.4534,0.5666]
B)[0.4625,0.5575]
C)[0.5084,0.5116]
D)[0.5086,0.5114]

Accepted Answer

The answer of Construct a 95% confidence interval on the...

Question 7

Exhibit 8-5.A sample of 2,007 American adults was asked how they viewed China,with 17% of respondents indicating "unfriendly" and 6% of respondents indicating "an enemy" (Gallup,Nov.30 - Dec.18,2011). Refer to Exhibit 8-5.Construct a 99% confidence interval of the proportion of American adults who view China as "an enemy."
A)[0.0463,0.0737]
B)[0.0476,0.0724]
C)[0.0512,0.0688]
D)[0.0597,0.0603]

Accepted Answer

The answer of Exhibit 8-5.A sample of 2,007 American adults...

Question 8

A medical devices company wants to know the number of hours their MRI machines are used per day.A previous study found a standard deviation of four hours.How many MRI machines must the company find data for in order to have a margin of error of at most 0.5 hours when calculating a 98% confidence interval?
A)346.26
B)347
C)424.69
D)425

Accepted Answer

The answer of A medical devices company wants to know...

Question 9

Which of the following is the correct formula for the margin of error in the interval estimation of p?
A) 
B) 
C) 
D)

Accepted Answer

The correct formula for the margin of error in interval estimation of p (proportion) is:

Margin of Error = z* × √ ( p*(1-p) / n )

where z* is the z-score corresponding to the desired level of confidence, p* is the estimated proportion, and n is the sample size.

Option B represents the correct formula with the correct characters in the equation.

Question 10

Exhibit 8-6.A sample of 1400 American households was asked if they planned to buy a new car next year.Of the respondents,34% indicated they planned to buy a new car next year. Refer to Exhibit 8-6.Construct a 90% confidence interval of the proportion of American households who expect to buy a new car next year.
A)[0.3152,0.3648]
B)[0.3191,0.3609]
C)[0.3394,0.3406]
D)[0.3354,0.3446]

Accepted Answer

To construct the confidence interval, we can use the formula:
Margin of error = z*sqrt((p̂*(1-p̂))/n)
where z is the z-score for the desired confidence level (90% corresponds to a z-score of 1.645), p̂ is the sample proportion (0.34), and n is the sample size (1400).
Plugging in the values, we get:
Margin of error = 1.645*sqrt((0.34*(1-0.34))/1400) = 0.0209
To find the lower and upper bounds of the interval, we can subtract and add the margin of error to the sample proportion:
Lower bound = 0.34 - 0.0209 = 0.3191
Upper bound = 0.34 + 0.0209 = 0.3609
Thus, the 90% confidence interval is [0.3191, 0.3609], which corresponds to choice B.

Question 11

The minimum sample size n required to estimate a population mean with 95% confidence and the desired margin of error 1.5 was found to be 198.Which of the following is the approximate value of the assumed estimate of the population standard deviation?
A)10.7688
B)12.8309
C)115.9671
D)164.6326

Accepted Answer

The answer of The minimum sample size n required to...

Question 12

Suppose we wish to find the required sample size to find a 90% confidence interval for the population proportion with the desired margin of error.If there is no rough estimate  of the population proportion,what value should be assumed for  ?
A)0.05
B)0.10
C)0.50
D)0.90

Accepted Answer

The answer of Suppose we wish to find the required...

Question 13

A marketing firm wants to estimate how much root beer the average teenager drinks per year.A previous study found a standard deviation of 1.12 liters.How many teenagers must the firm interview in order to have a margin of error of at most 0.1 liters when constructing a 99% confidence interval?
A)29
B)832
C)832.39
D)833

Accepted Answer

To find the sample size, use the formula for the margin of error in a confidence interval: $ E = Z \times \frac{\sigma}{\sqrt{n}} $. For a 99% confidence interval, the Z-value is approximately 2.576. Rearrange the formula to solve for $ n $: $ n = \left(\frac{Z \times \sigma}{E}\right)^2 $. Plugging in the values: $ n = \left(\frac{2.576 \times 1.12}{0.1}\right)^2 \approx 832.39 $. Since the sample size must be a whole number, round up to 833.

Question 14

When examining the possible outcome of an election,what type of confidence interval is most suitable for estimating the current support for a candidate?
A)The confidence interval for the sample mean
B)The confidence interval for the population mean
C)The confidence interval for the sample proportion
D)The confidence interval for the population proportion

Accepted Answer

The answer of When examining the possible outcome of an...

Question 15

When the required sample size calculated by using a formula is not a whole number,what is the best choice for the required sample size?
A)Round the result of the calculation in a standard way.
B)Round the result of the calculation up to the nearest whole number.
C)Round the result of the calculation down to the nearest whole number.
D)None of the above.

Accepted Answer

When the required sample size is not a whole number, it is best to round up to the nearest whole number to ensure that the sample size is not too small, which could lead to inaccurate results. Rounding down could result in a sample size that is too small, while rounding in a standard way could be ambiguous and not necessarily provide a conservative estimate of the necessary sample size. Therefore, rounding up to the nearest whole number is the best choice.

Question 16

The sampling distribution of the population proportion is based on a binomial distribution.What condition must be met to use the normal approximation for the confidence interval?
A)n ≥ 30
B) 
C) 
D)

Accepted Answer

The answer of The sampling distribution of the population proportion...

Question 17

Construct a 90% confidence interval on the population proportion for the support of candidate A in the following mayoral election.Candidate A is facing two opposing candidates.In a recent poll of 300 residents,153 supported her.
A)[0.4534,0.5666]
B)[0.4625,0.5575]
C)[0.5086,0.5114]
D)[0.5090,0.5110]

Accepted Answer

The sample proportion is 153/300 = 0.51. Using the formula for a confidence interval for a proportion, with a 90% confidence level, the margin of error is calculated using the standard error and the z-score for 90% confidence (approximately 1.645). The interval is [0.4625, 0.5575].

Question 18

What is the minimum sample size required to estimate a population mean with 90% confidence when the desired margin of error is D = 1.25? The standard deviation in a preselected sample is 8.5.
A)n = 76
B)n = 125
C)n = 126
D)n = 190

Accepted Answer

The formula for calculating the minimum sample size is:

n = (z-score)^2 * (standard deviation^2) / (margin of error)^2

Where the z-score for a 90% confidence interval is 1.645.

Plugging in the values:

n = (1.645)^2 * (8.5)^2 / (1.25)^2 = 126.07

Rounding up to the nearest integer, the minimum sample size required is 126. Therefore, the best choice is C, n = 126.

Question 19

What is the minimum sample size required to estimate a population mean with 95% confidence when the desired margin of error is D = 1.5? The population standard deviation is known to be 10.75.
A)n = 138
B)n = 139
C)n = 197
D)n = 198

Accepted Answer

The formula for calculating the minimum sample size required to estimate a population mean with a margin of error is:

n = (z^2 * σ^2) / D^2

Where z is the z-score for the desired level of confidence, σ is the population standard deviation, and D is the desired margin of error.

For a 95% confidence level, the z-score is 1.96. Plugging in the given values, we get:

n = (1.96^2 * 10.75^2) / 1.5^2 = 198.35

Rounding up to the nearest whole number, the minimum sample size required is 198. Therefore, the correct answer is D.

Question 20

Exhibit 8.4.According to a report in USA Today (February 1,2012),more and more parents are helping their young adult children get homes.Suppose 8 persons in a random sample of 40 young adults who recently purchased a home in Kentucky got help from their parents.You have been asked to construct a 95% confidence interval for the population proportion of all young adults in Kentucky who got help from their parents. Refer to Exhibit 8.4.If a 99% confidence interval is constructed instead of a 95% confidence interval for the population proportion,then ____________________.
A)The resulting margin of error will increase and the risk of reporting an incorrect interval will increase
B)The resulting margin of error will decrease and the risk of reporting an incorrect interval will increase
C)The resulting margin of error will increase and the risk of reporting an incorrect interval will decrease
D)The resulting margin of error will decrease and the risk of reporting an incorrect interval will decrease

Accepted Answer

Increasing the confidence level from 95% to 99% results in a wider confidence interval, which increases the margin of error but decreases the risk of reporting an incorrect interval.

Quiz 8: Interval Estimation

Statisticians like precision in their interval estimates.A low margin of error is needed to achieve this.Which of the following supports this when selecting sample sizes?

Construct a 95% confidence interval on the population proportion for the support of candidate A in the following mayoral election.Candidate A is facing two opposing candidates in a mayoral election.In a recent poll of 300 residents,she has garnered 51% support.

Which of the following is the correct formula for the margin of error in the interval estimation of p?

The minimum sample size n required to estimate a population mean with 95% confidence and the desired margin of error 1.5 was found to be 198.Which of the following is the approximate value of the assumed estimate of the population standard deviation?

Suppose we wish to find the required sample size to find a 90% confidence interval for the population proportion with the desired margin of error.If there is no rough estimate of the population proportion,what value should be assumed for ?

A marketing firm wants to estimate how much root beer the average teenager drinks per year.A previous study found a standard deviation of 1.12 liters.How many teenagers must the firm interview in order to have a margin of error of at most 0.1 liters when constructing a 99% confidence interval?

When examining the possible outcome of an election,what type of confidence interval is most suitable for estimating the current support for a candidate?

When the required sample size calculated by using a formula is not a whole number,what is the best choice for the required sample size?

The sampling distribution of the population proportion is based on a binomial distribution.What condition must be met to use the normal approximation for the confidence interval?

Construct a 90% confidence interval on the population proportion for the support of candidate A in the following mayoral election.Candidate A is facing two opposing candidates.In a recent poll of 300 residents,153 supported her.

What is the minimum sample size required to estimate a population mean with 90% confidence when the desired margin of error is D = 1.25? The standard deviation in a preselected sample is 8.5.

What is the minimum sample size required to estimate a population mean with 95% confidence when the desired margin of error is D = 1.5? The population standard deviation is known to be 10.75.