business stats unit 3

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.

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 greater, 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 bigger, the interval is narrower.

In general, the null and alternative hypotheses are __________.

mutually exclusive

If the p-value for a hypothesis test is 0.027 and the chosen level of significance is α = 0.05, then the correct conclusion is to __________.

reject the null hypothesis

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 If we choose a 5% significance level, we __________.

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

If the underlying populations cannot be assumed to be normal, then by the central limit theorem, the sampling distribution of X¯¯¯1 − X¯¯¯2�¯1 − �¯2 is approximately normal only if both sample sizes are sufficiently large—that is, when _________ .

n1 ≥ 30 and n2 ≥ 30

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

no error

If the p-value for a hypothesis test is 0.07 and the chosen level of significance is α = 0.05, then the correct conclusion is to __________.

not reject the null hypothesis

Selection bias occurs when ______.

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

When conducting a hypothesis test for a given sample size, if the probability of a Type I error decreases, then the __________.

probability of incorrectly not rejecting the null hypothesis increases

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

Which of the following is considered an estimate?

x¯= 20

If a population is known to be normally distributed, what can be said of 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.

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

1.645

What is zα / 2 ��⁢ / 2⁢ for a 95% confidence interval of the population mean?

1.96

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?

1.96(3.20/6)

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

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.

550 ± 1.96(120/5)

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.

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 μD.

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 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 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.

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.

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.

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?

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.

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

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

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; a different 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 μD 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, HA: µ1 − µ2 ≠ 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, HA: µ1 − µ2 > 0

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.

H0: µD = 0, HA: µD ≠ 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.

H0: µD = 0, HA: µD ≠ 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 µD 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: µD ≤ 50, HA: µD > 50

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 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 next. Use pretest score as µ1 for population 1 and posttest score as µ2 for population 2, or µD as the mean of the difference calculated as pretest score minus posttest score.

H0: µD ≥ 0, HA: µD < 0

Which of the following set of 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: µD ≥ 0,HA: µD < 0

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

Which of the following is not a form of bias?

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

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.

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?

John selects the first 25 homes that he passes as he walks into the entrance of the development.

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

Matched-pairs sampling

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

Normality of the estimator

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

Point estimate ± Margin of error

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

Reject H0 if the p-value < α.

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

Rejecting the null hypothesis when it is true.

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.

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 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.

How do the tdf and z distributions differ?

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

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.

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.

When calculating the standard error of X¯¯¯1−X¯¯¯2�¯1−�¯2 , under what assumption do you pool the sample variances s22�22 and s22�22 ?

Unknown population variances that are assumed equal.

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.

Which of the following is considered an estimator?

X¯¯¯

Consider the following hypotheses that relate to the medical field: H0: A person is free of disease. HA: A person has disease. In this instance, a Type I error is often referred to as __________.

a false positive

The tdf distribution is similar to the z distribution because __________.

as the degrees of freedom go to 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

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

The alternative hypothesis typically __________.

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

When comparing two population means, their hypothesized difference _________ .

may assume any value

For a given sample size n, __________.

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

A Type II error occurs when we __________.

do not reject the null hypothesis when it is actually false

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

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

is closer to a normal distribution

A Type I error occurs when we __________.

reject the null hypothesis when it is actually true

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 HA: μ < 6, μ stand for __________.

the mean delivery time

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. The population parameter to be tested is __________.

the mean number of customers visiting the dealership per day

If the null hypothesis is rejected at a 1% significance level, then __________.

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

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

the population variances are known

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

According to the central limit theorem, the distribution of the sample means is normal if ________.

the sample size n ≥ 30

A hypothesis test regarding the population mean is based on __________.

the sampling distribution of the sample mean

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

Bias can occur in sampling. Bias refers to ______.

the tendency of a sample statistic to systematically over- or underestimate a population parameter

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

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

Nonresponse bias occurs when ______.

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