Question 1

A certain population is strongly skewed to the right. We want to estimate its mean, so we will collect a sample. Which should be true if we use a large sample rather than a small one?
I. The distribution of our sample data will be closer to normal.
II. The sampling model of the sample means will be closer to normal. III. The variability of the sample means will be greater.

Accepted Answer

II. According to the Central Limit Theorem, as the sample size increases, the distribution of the sample means will approach a normal distribution, regardless of the shape of the population distribution.

Question 2

We have calculated a 95% confidence interval and would prefer for our next confidence interval to have a smaller margin of error without losing any confidence. In order to do this, we can
I. change the z* value to a smaller number. II. take a larger sample.
III. take a smaller sample.

Accepted Answer

Taking a larger sample (option II) will not necessarily lead to a smaller margin of error if the population standard deviation remains the same. Changing the z* value (option I) would require changing the confidence level and could result in a loss of confidence. Therefore, the best choice is to keep the same confidence level and reduce the margin of error by changing the sample size (option III). Taking a larger sample will result in a smaller margin of error, assuming all other factors are constant.

Question 3

A state's Department of Education reports that 12% of the high school students in that state attend private high schools. The State University wonders if the percentage is the same in their applicant pool. Admissions officers plan to check a random sample of the over 10,000 applications on file to estimate the percentage of students applying for admission who attend private schools.
-Should the admissions officers conclude that the percentage of private school students in their applicant pool is lower than the statewide enrollment rate of 12%? Explain.

Accepted Answer

The answer of A state's Department of Education reports that...

Question 4

Which of the following is true about Type I and Type II errors?
I. Type I errors are always worse than Type II errors.
II. The severity of Type I and Type II errors depends on the situation being tested.
III. In any given situation, the higher the risk of Type I error, the lower the risk of Type II error.

Accepted Answer

Statement I is not always true as the severity of the errors depends on the context. Sometimes, Type II errors can be severe, for example, in medical testing where a false negative result can mean a missed diagnosis leading to serious health consequences. Statement II is true as the severity of the errors depends on the specific situation being tested. Statement III is also true as increasing the threshold for Type I error (i.e., decreasing alpha value) reduces the probability of committing a Type I error but increases the probability of committing a Type II error (i.e., decreasing beta value). Therefore, the best choice is D.

Question 5

We have calculated a confidence interval based on a sample of n = 180. Now we want to get a better estimate with a margin of error only one third as large. We need a new sample with n at least…

Accepted Answer

Since we want the margin of error to be only one third as large, we need to increase the sample size by a factor of 9. This means that the new sample size would be 9 times greater than the original sample size, or n = 180 x 9 = 1,620. Therefore, the best choice is D.

Question 6

They actually select a random sample of 450 applications, and find that 46 of those students attend private schools. Create the confidence interval.

Accepted Answer

The answer of They actually select a random sample of...

Question 7

When you construct a 90% confidence interval for a population proportion based on a sample, what is it that you are pretty sure is in your interval?
I. 90% of the sample data.
II. 90% of all sample proportions. III. The population proportion.

Accepted Answer

When you construct a 90% confidence interval for a population proportion based on a sample, the interval is designed to capture the true population proportion with a 90% degree of confidence. Therefore, we are pretty sure that the true population proportion lies within the interval. This means that the correct answer is E, as the interval estimates the value of the population proportion, not the sample data or sample proportions themselves.

Question 8

Pew Research reports that 63% of the U.S. adult cell phone owners use their phone to go online. A company wants to target 16- to 24-year olds for advertising and they wonder if that age group has a similar pattern of phone use.
-Should the company conclude that the percentage of cell phone owners in this age group who use their phone to go online is different from 63%? Explain.

Accepted Answer

The answer of Pew Research reports that 63% of the...

Question 9

The board of directors for Procter and Gamble is concerned that only 19.5% of the people who use toothpaste buy Crest toothpaste. A marketing director suggests that the company invest in a new marketing campaign which will include advertisements and new labeling for the toothpaste. The research department conducts product trials in test markets for one month to determine if the market share increases with new labels.
-Over the trial month the market share in the sample rose to 22% of shoppers. The company's board of directors decided this increase was significant. Now that they have concluded the new marketing campaign works, why might they still choose not to invest in the campaign?

Accepted Answer

The answer of The board of directors for Procter and...

Question 10

Which of the following would increase the power of a test?
I. Increase the sample size
II. Decrease the significance level
III. Increase alpha

Accepted Answer

Increasing the sample size and increasing alpha (which means increasing the probability of making a Type I error) would both increase the power of a test. Decreasing the significance level (which means decreasing the probability of making a Type I error) would decrease the power of the test.

Question 11

A company manufacturing computer chips finds that 8% of all chips manufactured are defective. Management is concerned that employee inattention is partially responsible for the high defect rate. In an effort to decrease the percentage of defective chips, management decides to offer incentives to employees who have lower defect rates on their shifts. The incentive program is instituted for one month. If successful, the company will continue with the incentive program.
-Over the trial month, 6% of the computer chips manufactured were defective. Management decided that this decrease was significant. Why might management might choose not to permanently institute the employee incentive program?

Accepted Answer

The answer of A company manufacturing computer chips finds that...

Question 12

We have calculated a confidence interval based upon a sample of n = 200. Now we want to get a better estimate with a margin of error only one fifth as large. We need a new sample with n at least
…

Accepted Answer

The margin of error is inversely proportional to the square root of the sample size (n). To reduce the margin of error by a factor of 1/5, we need to increase the sample size by a factor of (5)^2 = 25. Therefore, the new sample size should be 200 * 25 = 5000.

Question 13

The manager of an orchard expects about 70% of his apples to exceed the weight requirement for "Grade A" designation. At least how many apples must he sample to be 90% confident of estimating the true proportion within ± 4%?

Accepted Answer

To determine the sample size needed for a proportion with a specific confidence level and margin of error, we use the formula: n = (Z^2 * p * (1-p)) / E^2. For 90% confidence, Z is approximately 1.645. Here, p = 0.7 and E = 0.04. Plugging these values into the formula gives approximately 356.

Question 14

Which is true about a 95% confidence interval based on a given sample?
I. The interval contains 95% of the population.
II. Results from 95% of all samples will lie in the interval.
III. The interval is narrower than a 98% confidence interval would be.

Accepted Answer

A 95% confidence interval is narrower than a 98% confidence interval because a higher confidence level requires a wider interval to ensure that the parameter is captured.

Question 15

According to the 2010 U.S. Census, 11.7% of the people in the state of Oregon were Hispanic or Latino. A political party wants to know how much impact the Hispanic and Latino vote will have, so they wonder if that percentage has changed since then. They take a random sample of 853 adults in Oregon and ask, among other things, their race. 113 of the people surveyed were Hispanic or Latino. Can the political party conclude that the Hispanic proportion of the population has changed since 2010?
-Was your test one-tail upper tail, lower tail, or two-tail? Explain why you chose that kind of test in this situation.

Accepted Answer

The answer of According to the 2010 U.S. Census, 11.7%...

Question 16

A certain population is bimodal. We want to estimate its mean, so we will collect a sample. Which should be true if we use a large sample rather than a small one?
I. The distribution of our sample data will be clearly bimodal.
II. The sampling distribution of the sample means will be approximately normal. III. The variability of the sample means will be smaller.

Accepted Answer

The answer of A certain population is bimodal. We want...

Question 17

Which is true about a 99% confidence interval based on a given sample?
I. The interval contains 99% of the population.
II. Results from 99% of all samples will lie in this interval.
III. The interval is wider than a 95% confidence interval would be.

Accepted Answer

III only. A 99% confidence interval does not imply that it contains 99% of the population or that 99% of all samples will lie within this interval. It means that if we were to take many samples and construct a confidence interval from each, 99% of those intervals would contain the true population parameter. A 99% confidence interval is wider than a 95% confidence interval because increasing confidence requires capturing more of the population variability to be more certain that the interval includes the true parameter.

Question 18

Which is true about a 98% confidence interval for a population proportion based on a given sample?
I. We are 98% confident that other sample proportions will be in our interval. II. There is a 98% chance that our interval contains the population proportion. III. The interval is wider than a 95% confidence interval would be.

Accepted Answer

III is correct because increasing the confidence level from 95% to 98% requires widening the interval to capture the population proportion with higher certainty. Statements I and II reflect common misconceptions; confidence intervals provide a range for the population parameter, not the probability of sample statistics falling within that range or the chance of the interval containing the parameter.

Question 19

We have calculated a confidence interval based on a sample of size n = 100. Now we want to get a better estimate with a margin of error that is only one-fourth as large. How large does our new sample need to be?

Accepted Answer

To reduce the margin of error to one-fourth, the sample size must be increased by a factor of 16 (since the margin of error is inversely proportional to the square root of the sample size). Therefore, the new sample size should be 100 * 16 = 1600.

Question 20

The board of directors for Procter and Gamble is concerned that only 19.5% of the people who use toothpaste buy Crest toothpaste. A marketing director suggests that the company invest in a new marketing campaign which will include advertisements and new labeling for the toothpaste. The research department conducts product trials in test markets for one month to determine if the market share increases with new labels.
-What level of significance did the research department use?

Accepted Answer

The answer of The board of directors for Procter and...

Quiz 5: Understanding and Comparing Distributions

We have calculated a 95% confidence interval and would prefer for our next confidence interval to have a smaller margin of error without losing any confidence. In order to do this, we can I. change the z* value to a smaller number. II. take a larger sample. III. take a smaller sample.

We have calculated a confidence interval based on a sample of n = 180. Now we want to get a better estimate with a margin of error only one third as large. We need a new sample with n at least…

They actually select a random sample of 450 applications, and find that 46 of those students attend private schools. Create the confidence interval.

When you construct a 90% confidence interval for a population proportion based on a sample, what is it that you are pretty sure is in your interval? I. 90% of the sample data. II. 90% of all sample proportions. III. The population proportion.

Which of the following would increase the power of a test? I. Increase the sample size II. Decrease the significance level III. Increase alpha

We have calculated a confidence interval based upon a sample of n = 200. Now we want to get a better estimate with a margin of error only one fifth as large. We need a new sample with n at least …

The manager of an orchard expects about 70% of his apples to exceed the weight requirement for "Grade A" designation. At least how many apples must he sample to be 90% confident of estimating the true proportion within ± 4%?

Which is true about a 95% confidence interval based on a given sample? I. The interval contains 95% of the population. II. Results from 95% of all samples will lie in the interval. III. The interval is narrower than a 98% confidence interval would be.

Which is true about a 99% confidence interval based on a given sample? I. The interval contains 99% of the population. II. Results from 99% of all samples will lie in this interval. III. The interval is wider than a 95% confidence interval would be.

We have calculated a confidence interval based on a sample of size n = 100. Now we want to get a better estimate with a margin of error that is only one-fourth as large. How large does our new sample need to be?