Business Stats Unit 3

Bias can occur in sampling. Bias refers to __________

the tendency of a sample statistic to systematically overestimate or underestimate a population parameter.

Selection bias occurs when _________

portions of the population are underrepresented or excluded from the consideration for the sample

Nonresponse bias occurs when __________

those responding to a survey or poll differ systematically from the nonrespondents

Which of the following is not a form of bias?

Information from the sample is typical of information in the population.

Which of the following meets the requirements of a simple random sample?

A population contains 10 members under the age of 25 and 20 members over the age of 25. The sample will include six people chosen at random, without regard to age.

Which of the following meets the requirements of a cluster sample?

A population can be divided into 50 city blocks. The sample will include residents from two randomly chosen city blocks.

Which of the following meets the requirements of a stratified random sample?

A population contains 10 members under the age of 25 and 20 members over the age of 25. The sample will include two people chosen at random under the age of 25 and four people chosen at random over 25.

Which of the following is true about statistics such as the sample mean or sample proportion?

A statistic is a random variable.

Statistics are used to estimate population parameters, particularly when it is impossible or too expensive to poll an entire population. A particular value of a statistic is referred to as a(n) __________

estimate

Which of the following is considered an estimator?

X bar

Which of the following is considered an estimate?

x bar = 20

How does the variance of the sample mean compare to the variance of the population?

It is smaller and therefore suggests that averages have less variation than individual observations.

If a population is known to be normally distributed, what can be said about the sampling distribution of the sample mean drawn from this population?

For any sample size n, the sampling distribution of the sample mean is normally distributed.

Over the entire six years that students attend an Ohio elementary school, they are absent, on average, 28 days due to influenza. Assume that the standard deviation over this time period is σ = 9 days. Upon graduation from elementary school, a random sample of 36 students is taken and asked how many days of school they missed due to influenza. What is the expected value for the sampling distribution of the average number of school days missed due to influenza?

28

According to the central limit theorem, the sampling distribution of the sample means is approximately normal if _________

the sample size n ≥ 30

The central limit theorem states that, for any distribution, as the sample size n gets larger, the sampling distribution of the sample mean __________

gets closer to a normal distribution.

John would like to conduct a survey in his neighborhood to get homeowners’ opinion on the Delmarva proposal to switch to natural gas. Which of the following is an example of a stratified sample?

John divides the population into two-story, split-level, and ranch houses. Then, he selects a proportional number of houses from each group.

Peggy would like to conduct a survey in her neighborhood to get homeowners’ opinions on the Delmarva proposal to switch to natural gas. Which of the following is an example of a convenience sample?

Peggy selects the first 25 homes that she passes as she walks into the entrance of the development.

A survey is designed to collect data on students’ evaluations of their instructor’s teaching performance. Which of the following situations most likely results in social-desirability biases?

The survey is administered in class with the instructor in the room the entire time.

Which of the following statements regarding the sampling distribution of the sample mean is TRUE?

The variance of the sample mean is smaller than the variance of all individual observations in the population.

What is the purpose of calculating a confidence interval?

To provide a range of values that, with a certain measure of confidence, contains the population parameter of interest.

What is the most typical form of a calculated confidence interval?

Point estimate ± Margin of error

Which of the following is the necessary condition for creating confidence intervals for the population mean?

Normality of the estimator

What is za/2 for a 95% confidence interval of the population mean?

1.96

What is za/2 for a 90% confidence interval of the population mean?

1.645

The daily revenue from the sale of fried dough at a local street vendor in Boston is known to be normally distributed with a known standard deviation of $120. The revenue on each of the last 25 days is noted, and the average is computed as $550. Construct a 95% confidence interval for the population mean of the sale of fried dough by this vendor.

[502.9609, 597.0391]

The ages of MBA students at a university are normally distributed with a known population variance of 10.24. Suppose you are asked to construct a 95% confidence interval for the population mean age if the mean of a sample of 36 students is 26.5 years. What is the margin of error for a 95% confidence interval for the population mean?

1.0453

The ages of MBA students at a university are normally distributed with a known population variance of 10.24. Suppose you are asked to construct a 95% confidence interval for the population mean age if the mean of a sample of 36 students is 26.5 years. If a 99% confidence interval is constructed instead of a 95% confidence interval for the population mean, then __________

the resulting margin of error will increase and the risk of reporting an incorrect interval will decrease

An analyst takes a random sample of 25 firms in the telecommunications industry and constructs a confidence interval for the mean return for the prior year. Holding all else constant, if he increased the sample size to 30 firms, how are the standard error of the mean and the width of the confidence interval affected?

The standard error of the mean gets smaller, and the confidence interval gets narrower.

For a given confidence level and sample size, which of the following is true in the interval estimation of the population mean when σ is known?

If the population standard deviation is larger, the interval is wider.

For a given confidence level and population standard deviation, which of the following is true in the interval estimation of the population mean?

If the sample size is larger, the interval is narrower.

For a given sample size and population standard deviation, which of the following is true in the interval estimation of the population mean?

If the confidence level is greater, the interval is wider.

A 90% confidence interval is constructed for the population mean. If a 95% confidence interval had been constructed instead (everything else remaining the same), the width of the interval would have been __________ and the probability of making an error would have been __________

wider; smaller

The t_df  distribution is similar to the z distribution because __________

as the degrees of freedom approaches infinity, the t distribution converges to the z distribution and both have asymptotic tails—that is, their tails become closer and closer to the horizontal axis but never touch it

How do the t_df and z distributions differ?

The t_df distribution has broader tails (it is flatter around zero).

Confidence intervals of the population mean may be created for the cases when the population standard deviation is known or unknown. How are these two cases treated differently?

Use the z table when σ is known; use the t table when σ is unknown.

What conditions are required by the central limit theorem before a confidence interval of the population mean may be created?

The underlying population need not be normally distributed if the sample size is 30 or more.

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 larger sample size reduces the margin of error.

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?

Round the result of the calculation up to the nearest whole number.

A machine that is programmed to package 1.20 pounds of cereal is being tested for its accuracy in a sample of 36 cereal boxes: the sample mean filling weight is calculated as 1.22 pounds. The population standard deviation is known to be 0.06 pounds. Which of the following conclusions is correct with a 95% confidence level?

The machine is operating properly because the interval contains the target.

In general, the null and alternative hypotheses are _________

mutually exclusive

The alternative hypothesis typically __________

contests the status quo, for which a corrective action may be required.

The national average for an eighth grade reading comprehension test is 73. A school district claims that its eighth graders outperform the national average. In testing the school district’s claim, how does one define the population parameter of interest?

The mean score on the eighth grade reading comprehension test

A local courier service advertises that its average delivery time is less than 6 hours for local deliveries. When testing the two hypotheses, H₀: μ ≥ 6 and H_A: μ < 6, μ stand for __________

the mean delivery time

A Type I error occurs when we

reject the null hypothesis when the null hypothesis is actually true.

A Type II error occurs when we __________

do not reject the null hypothesis when the null hypothesis is actually false

When we reject the null hypothesis when it is actually false, we have committed __________

no error

For a given sample size n, which of the following statements is correct?

decreasing the probability of a Type I error α will increase the probability of a Type II error β

Consider the following hypotheses that relate to the medical field:

H₀: A person is free of disease.

H_A: A person has a disease.

In this instance, a Type I error is often referred to as __________

a false positive

When conducting a hypothesis test, which of the following decisions represents an error?

Rejecting the null hypothesis when it is true.

A hypothesis test regarding the population mean is based on __________

the sampling distribution of the sample mean

If the chosen significance level is α = 0.05, then __________

there is a 5% probability of rejecting a true null hypothesis.

For a given sample size, when conducting a hypothesis test for population mean or proportion, if the probability of a Type Ione error decreases, then the _________

probability of incorrectly not rejecting the null hypothesis increases.

If the p-value for a hypothesis test is 0.027 and the chosen level of significance is α = 0.05, what is the correct conclusion?

reject the null hypothesis.

If the p-value for a hypothesis test is 0.07 and the chosen level of significance is α = 0.05, what is the correct conclusion?

do not reject the null hypothesis.

If the null hypothesis is rejected at a 1% significance level, then which of the following statements is correct?

the null hypothesis will be rejected at a 5% significance level.

What is the decision rule when using the p-value approach to hypothesis testing?

Reject H₀ if the p-value < α.

The owner of a large car dealership believes that the financial crisis decreased the number of customers visiting her dealership. The dealership has historically had 800 customers per day. The owner takes a sample of 100 days and finds the average number of customers visiting the dealership per day was 750. Assume that the population standard deviation is 350. What is the population parameter to be tested?

the mean number of customers visiting the dealership per day

A 99% confidence interval for the population mean when the population standard deviation is known yields the following interval: [−3.79, 5.86]. At the 1% significance level, what decision should be made regarding the following hypothesis test with H₀:μ = 0,H_A:μ ≠ 0?

Do not reject H₀; we cannot conclude that the mean differs from zero

To test if the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the value of the test statistic is computed as t₂₉ = 2.42. What is the correct decision at the 5% significance level?

reject the null hypothesis and conclude that the mean IQ is greater than 100.

Two or more random samples are considered independent if_______.

the process that generates one sample is completely separate from the process that generates the other sample(s).

The choice of an appropriate test for comparing two population means depends on whether we deal with the following except:

the sum or product of a single parameter

Which of the following is not a restriction for comparing two population means?

Matched-pairs sampling

When comparing two population means, their hypothesized difference __________

may assume any value

Suppose you want to perform a test to compare the mean GPA of all freshmen with the mean GPA of all sophomores in a college. What type of sampling is required for this test?

Independent sampling with numerical data

If the underlying populations cannot be assumed to be normal, then by the central limit theorem, the sampling distribution X₁ bar - X₂ bar of is approximately normal only if both sample sizes are sufficiently large—that is, when __________

n₁ ≥ 30 and n₂ ≥ 30

When calculating the standard error of X₁ bar - X₂ bar, under what assumption do you pool the sample variances s 2/2 and s 2/2?

Unknown population variances that are assumed equal.

When testing the difference between two population means under independent sampling, we use the z distribution if _________

the population variances are known.

What type of test for population means should be performed when employees are first tested, trained, and then retested?

A t test under dependent sampling.

A particular personal trainer works primarily with track and field athletes. She believes that her clients run faster after going through her program for six weeks. How might she test that claim?

A matched-pairs hypothesis test for μ_D.

What type of data is required to compare prices of the same textbooks sold by two different vendors?

Dependent random samples with numerical data.

Which of the following hypotheses is used to test if the mean of the first population is smaller than the mean of the second population, using matched-paired sampling?

H₀: µ_D ≥ 0,H_A: µ_D < 0

A farmer uses a lot of fertilizer to grow his crops. The farmer’s manager thinks fertilizer products from distributor A contain more of the nitrogen that his plants need than distributor B’s fertilizer does. He takes two independent samples of four batches of fertilizer from each distributor and measures the amount of nitrogen in each batch. Fertilizer from distributor A contained 23 pounds per batch and fertilizer from distributor B contained 18 pounds per batch. Suppose the population standard deviation for distributor A and distributor B is four pounds per batch and five pounds per batch, respectively. Assume the distribution of nitrogen in fertilizer is normally distributed. Let µ₁ and µ₂ represent the average amount of nitrogen per batch for fertilizer’s A and B, respectively. Specify the competing hypotheses to determine if fertilizer A contains more nitrogen per batch than fertilizer B.

H₀: µ₁ − µ₂ ≤ 0, H_A: µ₁ − µ₂ > 0

A 7,000-seat theater is interested in determining whether there is a difference in attendance between shows on Tuesday evening and those on Wednesday evening. A random sample of 25 weeks is collected for Tuesday; and a different random sample of 25 weeks is collected for Wednesday. The mean attendance on Tuesday evening is calculated as 5,500, while the mean attendance on Wednesday evening is calculated as 5,850. The known population standard deviation for attendance on Tuesday evening is 550 and the known population standard deviation for attendance on Wednesday evening is 445. Let μ₁ be the population mean of Tuesday, μ₂ be the population mean of Wednesday, and μ_D be the mean difference for a matched-pairs sampling. What are the appropriate hypotheses to determine whether there is a difference, on average, in attendance between shows on Tuesday evening and Wednesday evening?

H₀: µ₁ − µ₂ = 0, H_A: µ₁ − µ₂ ≠ 0

A new sales training program has been instituted at a rent-to-own company. Prior to the training, 10 employees were tested on their knowledge of products offered by the company. Once the training was completed, the same employees were tested again in an effort to determine whether the training program was effective. Scores are known to be normally distributed. The sample scores on the tests are listed below. Use pretest score as µ₁ for population 1 and posttest score as µ₂ for population 2, or µ_D as the mean of the difference calculated as pretest score minus post-test score.

Pretest Score	Post-test Score
66	75
94	100
87	93
84	85
76	75
88	90

Which of the following are the appropriate hypotheses to determine if the training increases scores?/

H₀: µ_D ≥ 0/H_A: µ_D < 0

A tutor promises to improve GMAT scores of students by more than 50 points after three lessons. To see if this is true, the tutor takes a sample of 49 students’ test scores after and before they received tutoring. The mean difference was 53 points better after tutoring, with a standard deviation of the difference equal to 12 points. Let µ_D denote the mean of the difference: score after tutoring minus score before tutoring. Which of the following hypotheses will determine if the students improved their test scores by more than 50 points after being tutored?

H₀: µ_D ≤ 50, H_A: µ_D > 50

A bank is trying to determine which model of safe to install. The bank manager believes that each model is equally resistant to safe crackers but sets up a test to be sure. He hires nine safe experts to break into each of the models, timing each endeavor. The results (in seconds) are given next, paired by expert. Let D be the difference: Time to break Safe 1 minus Time to break Safe 2.

Safe 1	Safe 2
103	101
90	94
64	58
120	112
104	103
92	90
145	140
106	110
76	74

Which of the following hypotheses will determine if the two safes take, on average, the same amount of time to crack?

H₀: µ_D = 0, H_A: µ_D ≠ 0

A farmer is concerned that a change in fertilizer to an organic variant might change his crop yield. He subdivides his six lots and uses the old fertilizer on one half of each lot and the new fertilizer on the other half. The following table shows the results.

Lot	Crop Yield Using Old Fertilizer	Crop Yield Using New Fertilizer
1	10	12
2	11	10
3	10	13
4	9	9
5	12	11
6	11	12

Which of the following are the appropriate competing hypotheses?

H₀: µ_D = 0, H_A: µ_D ≠ 0

Which of the following is an example about analyzing the difference between two population means?

Compare the average six-year graduation rate between private and public colleges.

Which of the following is NOT an example of analyzing the mean difference of two populations based on matched-pairs sampling?

Compare the mean wait time of customers being served at a bank before and after the weekend.