Q&A Hero

Q: NARRBEGIN: SA_65_66 A company employs two shifts of workers. Each shift produces a type of gasket where the thickness is the critical dimension. The average thickness and the standard deviation of thickness for shift 1, based on a random sample of 40 gaskets, are 10.85 mm and 0.16 mm, respectively. The similar figures for shift 2, based on a random sample of 30 gaskets, are 10.90 mm and 0.19 mm. Let be the difference in thickness between shifts 1 and 2, and assume that the population variances are equal. NARREND (A) Construct a 95% confidence interval for. (B) Based on your answer to (A), are you convinced that the gaskets from shift 2 are, on average, wider than those from shift 1? Why or why not?

Q: NARRBEGIN: SA_62_64Q-Mart is interested in comparing its male and female customers. Q-Mart would like to know if its female charge customers spend more money, on average, than its male charge customers. They have collected random samples of 25 female customers and 22 male customers. On average, women charge customers spend $102.23 and men charge customers spend $86.46. Some information are shown below.Summary statistics for two samples FemaleMaleSample sizes2522Sample means102.23 86.46Sample standard deviations93.39359.695 Confidence interval for difference between means Sample mean difference 15.77 Pooled standard deviation79.466 Std error of difference 23.23 NARREND(A) Use a t - value of 2.014 to calculate a 95% confidence interval for the difference between the average female purchase and the average male purchase. Would you conclude that there is a significant difference between females and males in this case? Explain.(B) What are the degrees of freedom for the t-multiple in this calculation? Explain how you would calculate the degrees of freedom in this case.(C) What is the assumption in this case that allows you to use the pooled standard deviation for this confidence interval?

Q: NARRBEGIN: SA_60_61 The employee benefits manager of a medium size business would like to estimate the proportion of full-time employees who prefer adopting plan A of three available health care plans in the coming annual enrollment period. A reliable frame of the company's employees and their tentative health care preferences are available. Using Excel, the manager chose a random sample of size 50 from the frame. There were 17 employees in the sample who preferred plan A. NARREND (A) Construct a 99% confidence interval for the proportion of company employees who prefer plan A. Assume that the population consists of the preferences of all employees in the frame. (B) Interpret the 99% confidence interval constructed in (A).

Q: NARRBEGIN: SA_57_59A marketing research consultant hired by Coca-Cola is interested in determining the proportion of customers who favor Coke over other soft drinks. A random sample of 400 consumers was selected from the market under investigation and showed that 53% favored Coca-Cola over other brands.NARREND(A) Compute a 95% confidence interval for the true proportion of people who favor Coke. Do the results of this poll convince you that a majority of people favors Coke?(B) Suppose 2,000 (not 400) people were polled and 53% favored Coke. Would you now be convinced that a majority of people favor Coke? Why might your answer be different than in (A)?(C) How many people would have to be surveyed to be 95% confident that you can estimate the fraction of people who favor Coca-Cola within 1%?

Q: You have been assigned to determine whether more people prefer Coke to Pepsi. Assume that roughly half the population prefers Coke and half prefers Pepsi. How large a sample would you need to take to ensure that you could estimate, with 95% confidence, the proportion of people preferring Coke within 3% of the actual value?

Q: You are trying to estimate the average amount a family spends on food during a year. In the past, the standard deviation of the amount a family has spent on food during a year has been approximately $1200. If you want to be 99% sure that you have estimated average family food expenditures within $60, how many families do you need to survey?

Q: You are told that a random sample of 150 people from Iowa has been given cholesterol tests, and 60 of these people had levels over the "safe" count of 200. Construct a 95% confidence interval for the population proportion of people in Iowa with cholesterol levels over 200.

Q: If we cannot make the strong assumption that the variances of two samples are equal, then we must use the pooled standard deviation in calculating the standard error of a difference between the means.

Q: If two samples contain the same number of observations, then the data must be paired.

Q: Samples of exam scores for employees before and after a training class would be examples of paired data

Q: The approximate standard error of the point estimate of the population total is .

Q: The confidence interval for the population standard deviation s is centered at the point estimate, the sample standard deviation s.

Q: If a sample has 20 observations and a 95% confidence estimate for is needed, the appropriate value of t-multiple is 2.093

Q: The mean of the sampling distribution of the sample proportion, when the sample size n = 100 and the population proportion p = 0.15, is 15.0.

Q: In determining the sample size n for estimating the population proportion p, a conservative value of n can be obtained by using 0.50 as an estimate of p.

Q: The t-distribution and the standard normal distribution are practically indistinguishable as the degrees of freedom increase.

Q: The upper limit of the 90% confidence interval for the population proportionp, given that n = 100; and = 0.20 is 0.2658.

Q: In general, increasing the confidence level will narrow the confidence interval, and decreasing the confidence level widens the interval.

Q: In developing confidence interval for the difference between two population means using two independent samples, we use the pooled estimate in estimating the standard error of the sampling distribution of the sample mean difference if the populations are normal with equal variances.

Q: When samples of size n are drawn from a population, then the sampling distribution of the sample mean is approximately normal, provided that n is reasonably large.

Q: A 90% confidence interval estimate for a population mean is determined to be 72.8 to 79.6. If the confidence level is reduced to 80%, the confidence interval for becomes narrower.

Q: We can form a confidence interval for the population total T by finding a confidence interval for the population mean in the usual way, and then multiplying the lower and upper limits the confidence interval by the population size N.

Q: In order to construct a confidence interval estimate of the population mean, the value of must be given.

Q: The 95% confidence interval for the population mean, given that the sample size n = 49 and the population standard deviation = 7, is.

Q: A confidence interval is an interval estimate for which there is a specified degree of certainty that the actual true value of the population parameter will fall within the interval.

Q: If two random samples of sizes 30 and 35 are selected independently from two populations whose means are 85 and 90, then the mean of the sampling distribution of the sample mean difference, , equals 5.

Q: The degrees of freedom for the t and chi-square distributions is a numerical parameter of the distribution that defines the precise shape of the distribution.

Q: If two random samples of size 40 each are selected independently from two populations whose variances are 35 and 45, then the standard error of the sampling distribution of the sample mean difference, , equals 1.4142.

Q: If a random sample of size 250 is taken from a population, where it is known that the population proportion p = 0.4, then the mean of the sampling distribution of the sample proportion is 0.60.

Q: The lower limit of the 95% confidence interval for the population proportion p, given that n = 300; and = 0.10 is 0.1339.

Q: If the standard error of the sampling distribution of the sample proportion is 0.0324 for samples of size 200, then the population proportion must be 0.30.

Q: In general, the paired-sample procedure is appropriate when the samples are naturally paired in some way and there is a reasonably large positive correlation between the pairs. In this case, the paired-sample procedure makes more efficient use of the data and generally results in narrower confidence intervals.

Q: As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if the sample size n is greater than 30.

Q: The interval estimate 18.52.5 was developed for a population mean when the sample standard deviation s was 7.5. Had s equaled 15, the interval estimate would be 375.0.

Q: In developing a confidence interval for the population standard deviation , we make use of the fact that the sampling distribution of the sample standard deviation s is not the normal distribution or the t-distribution, but rather a right-skewed distribution called the chi-square distribution, which (for this procedure) has n " 1 degrees of freedom.

Q: The standard error of the sampling distribution of the sample proportion,when the sample size n = 50 and the population proportion p = 0.25, is 0.00375.

Q: For a given confidence level, the procedure for controlling interval length usually begins with the specification of a. the point estimate b. the population standard deviation, s c. the sample standard deviation, s d. the interval half-length, B

Q: An example of a problem where the sample data would be paired is: a. Difference between the means of appraised and sales house prices b. Difference between the proportion of defective items from two suppliers c. Difference in the mean life of two major brands of batteries d. Difference in the mean salaries for graduates in two different academic fields at a university e. None of these options

Q: The chi-square distribution for developing a confidence interval for a standard deviation has_____ degrees of freedom. a. n + 2 b. n +1 c. n d. n " 1 e. n - 2

Q: The approximate standard error of the point estimate of the population total is: a. b. c. d.

Q: The general form of a confidence interval is: a. Point Estimate = Multiple Standard Error b. Point Estimate = Multiple +Standard Error c. Point Estimate Multiple Standard Error d. Point Estimate = Multiple Standard Error

Q: The number of degrees of freedom needed to construct 90% confidence interval for the difference between means when the data are gathered from paired samples, with 15 observations in each sample, is: a. 30 b. 15 c. 28 d. 14

Q: From a sample of 500 items, 30 were found to be defective. The point estimate of the population proportion defective will be: a. 0.06 b. 30.0 c. 16.667 d. None of the above

Q: After calculating the sample size needed to estimate a population proportion to within 0.05, you have been told that the maximum allowable error (B) must be reduced to just 0.025. If the original calculation led to a sample size of 1000, the sample size will now have to be: a. 2000 b. 4000 c. 1000 d. 8000

Q: Two independent samples of sizes 20 and 25 are randomly selected from two normal populations with equal variances. In order to test the difference between the population means, the test statistic is: a. a standard normal random variable b. approximately standard normal random variable c. t-distributed with 45 degrees of freedom d. t-distributed with 43 degrees of freedom

Q: Two independent samples of sizes 50 and 50 are randomly selected from two populations to test the difference between the population means, . The sampling distribution of the sample mean difference is: a. normally distributed b. approximately normal c. t - distributed with 98 degrees of freedom d. chi-squared distributed with 99 degrees of freedom

Q: A parameter such as is sometimes referred to as a ________ parameter, because many times we need its value even though it is not the parameter of primary interest. a. special b. random c. nuisance d. independent e. dependent

Q: When the samples we want to compare are paired in some natural way, such as pretest/posttest for each person or husband/wife pairs, a more appropriate form of analysis is to not compare two separate variables, but their _____. a. difference b. sum c. ratio d. total e. product

Q: As the sample size increases, the t-distribution becomes more similar to the ________ distribution. a. normal b. exponential c. multinominal d. chi-square e. binomial

Q: The t-distribution for developing a confidence interval for a mean has _____ degrees of freedom. a. n + 2 b. n +1 c. n d. n " 1 e. n - 2

Q: If the odds of a horse winning a race are 2 to 1, then the probability of this horse winning the race is_____. a. 1/4 b. 1/3 c. 1/2 d. 2/3 e. None of the above

Q: There are, generally speaking, two types of statistical inference. They are: a. sample estimation and population estimation b. confidence interval estimation and hypothesis testing c. interval estimation for a mean and point estimation for a proportion d. independent sample estimation and dependent sample estimation e. None of the above

Q: When we replace with the sample standard deviation (s), we introduce a new source of variability and the sampling distribution becomes the _____. a. t -distribution b. F- distribution c. chi-square distribution d. normal distribution

Q: Suppose there are 500 accounts in a population. You sample 50 of them and find a sample mean of $500. What would be your estimate for the population total? a. $5,000 b. $50,000 c. $250,000 d. $500,000 e. None of the above

Q: Confidence intervals are a function of which of the following three things? a. The population, the sample, and the standard deviation b. The sample, the variable of interest, and the degrees of freedom c. The data in the sample, the confidence level, and the sample size d. The sampling distribution, the confidence level, and the degrees of freedom e. The mean, median, and mode

Q: When you calculate the sample size for a proportion, you use an estimate for the population proportion; namely. A conservative value for n can be obtained by using = _____. a. 0.01 b. 0.05 c. 0.10 d. 0.50 e. 1.00

Q: If you are constructing a confidence interval for a single mean, the confidence interval will _____ with an increase in the sample size. a. decrease b. increase c. stay the same d. increase or decrease, depending on the sample data

Q: If you increase the confidence level, the confidence interval _____. a. decreases b. increases c. stays the same d. may increase or decrease, depending on the sample data

Q: The chi-square and F-distributions are used primarily to make inferences about population ___________. a. means b. variances c. medians d. modes e. proportions

Q: NARRBEGIN: SA_117_120A columnist for the LA Times is working to meet a deadline on a story about commuting in Los Angeles. She wants to include information about the current price of gasoline in the Los Angeles metro area, but her source person for this type of information has already gone home for the day. So she decides to take her own sample as she drives home, writing down the prices she observes as she makes her way from downtown to her neighborhood in the suburbs. Below is the data sample she obtains (units are $/gallon).NARREND(A) Do you think she has obtained a true random sample?(B) What average price could she report, based on the above sample?(C) What average price range could she report, based on the above sample?(D) Do you see any issues with reporting the range calculated for (C)?

Q: NARRBEGIN: SA_112_116 Suppose that the average weekly earnings for employees in general automotive repair shops is $450, and that the standard deviation for the weekly earnings for such employees is $50. A sample of 100 such employees is selected at random. NARREND (A) Find the mean and standard deviation of the sampling distribution of the average weekly earnings in the sample. (B) Find probability that the mean of the sample is less than $445. (C) Find the probability that the mean of the sample is between $445 and $455. (D) Find the probability that the mean of the sample is greater than $460. (E) Explain why the assumption of normality about the distribution of the average weekly earnings for employees was not involved in the answers to (A) through (D).

Q: NARRBEGIN: SA_110_111A university bookstore manager is mildly concerned about the number of textbooks that were under-ordered and thus unavailable two days after the beginning of classes. The manager instructs an employee to pick a random number, go to the place where that number book is shelved, examine the next 50 titles, and record how many titles are unavailable.NARREND(A) Technically, this process does not yield a random sample of the books in the store. Why not?(B) How could a truly random sample be obtained?

Q: NARRBEGIN: SA_106_109The manager of a local fast-food restaurant is interested in improving service provided to customers who use the restaurant's drive-up window. As a first step in the process, the manager asks his assistant to record the time (in minutes) it takes to serve a large number of customers at the final window in the facility's drive-up system. The given frame in this case is 200 customer service times observed during the busiest hour of the day for this fast-food restaurant. The frame of 200 service times yielded a mean of 0.881. A simple random sample of 10 from this frame is presented below.Customer12345678910Service time1.021.180.950.900.851.100.750.601.251.00NARREND(A) Compute the point estimate of the population mean from the sample above. What is the sampling error in this case? Assume that the population consists of the given 200 customer service times.(B) Compute the point estimate of the population standard deviation from the sample above.(C) Should you use the finite population correction (fpc) factor to estimate the standard error of? Explain. If your answer is yes, what is the value of the fpc?(D) Determine a good approximation to the standard error of the mean in this case.

Q: NARRBEGIN: SA_103_105Sally Bird of Big Rapids Realty has received data on 60 houses that were recently sold in Mecosta County in Michigan. The data are recorded in the table shown below. Included in this data set are observations for each of the following variables:The appraised value of each house (in thousands of dollars)The selling price of each house (in thousands of dollars)The size of each house (in hundreds of square feet)The number of bedrooms in each houseNARREND(A) Suppose that Sally wishes to examine a representative subset of these 60 houses that has been stratified by the number of bedrooms. Use Excel to assist her by finding such a stratified sample of size 10 with proportional sample sizes.(B) Explain how Sally could apply cluster sampling in selecting a sample of size 15 from this frame.(C) What are the advantages and disadvantages of employing cluster sampling in this case?

Q: NARRBEGIN: SA_96_102 The manager of a small computer company has collected current annual salaries and number of years of post-secondary education for 52 full-time employees. The data are shown below: Current annual salaries: Number of years of post-secondary education: NARREND (A) Compute the mean, median, and standard deviation of the annual salaries for the 52 employees in the given frame. (B) Use Excel to choose a systematic sample of size 13 from the frame of annual salaries. (C) Compute the mean, median, and standard deviation of the annual salaries for the 13 employees included in your systematic sample in (B) (D) Compare your statistics in (C) with your computed descriptive measures for the frame in (A). Is your systematic sample representative of the frame with respect to the annual salary variable? (E) Assume that we wish to stratify these employees by the number of years of post-secondary education, select such a stratified sample of size 15 with approximately proportional sample sizes. (F) Compute the mean, median, and standard deviation of the annual salaries for the 15 employees included in your stratified sample in (E). (G) Compare these statistics in (F) with your computed descriptive measures for the frame obtained in (A). Is your stratified sample representative of the frame with respect to the annual salary variable?

Q: NARRBEGIN: SA_94_95 Suppose that you are an entrepreneur interested in establishing a new Internet-based auction service. Furthermore, suppose that you have gathered basic demographic information on a large number of Internet users. You currently have information on 1000 individuals related to their gender, age, education, marital status, annual household income, and number of people in household. Assume that these individuals were carefully selected through stratified sampling. NARREND (A) To assess potential interest in your proposed enterprise, you would like to conduct telephone interviews with a representative subset of the 1000 Internet users. How would you proceed to stratify the given frame of 1000 individuals to choose 50 for telephone interviews? Explain your approach. (B) Explain how you could apply cluster sampling to obtain a sample size of 50 from this frame. What are the advantages and disadvantages of employing cluster sampling in this case?

Q: NARRBEGIN: SA_92_93An editor of a local newspaper is concerned with the number of errors that are found in the daily paper. In order to understand the extent of this problem, the editor would like to estimate the average number of errors in the daily paper. The frame in this case is the number of errors found in the daily paper for the past six months (180 issues).NARREND(A) What sample size would be required for the production personnel to be approximately 95% sure that their estimate of the average number of errors per issue is within 4 errors of the true mean? Assume that the editor's best estimate of the population standard deviation () is 10 errors per issue.(B) How does your answer to (A) change if the editor wants the estimate to be within 3 errors of the actual population mean? Explain the difference in your answers to (A) and (B).

Q: NARRBEGIN: SA_86_90A statistics professor has just given the final examination in his introductory statistics course. In particular, he is interested in learning how his class of 50 students performed on this exam. The data are shown below.7872737579727577717883847181827971738974759374888390827962738876767680848491707674688087928479809174NARRENDA cannery claims that its sardine cans have a net weight of 8 oz., with a standard deviation of 0.1 oz. You take a simple random sample of 30 cans and encounter a sample mean of 7.85 oz. Are you inclined to believe the claim?

Q: NARRBEGIN: SA_86_90A statistics professor has just given the final examination in his introductory statistics course. In particular, he is interested in learning how his class of 50 students performed on this exam. The data are shown below.7872737579727577717883847181827971738974759374888390827962738876767680848491707674688087928479809174NARREND(A) Using these 50 students as the frame, use Excel to generate a simple random sample of size 10 from this frame.(B) Compute the mean scores in the frame and the simple random sample you generated in (A).(C) Compare the mean scores you computed in (B). Is your simple random sample a good representative of the frame? Why or why not?(D) Using these 50 students as the frame, use Excel to generate a systematic sample of size 10 from this frame.(E) Compare the mean scores in the frame with that in the systematic sample in (D). What do you conclude?

Q: NARRBEGIN: SA_83_85Auditors of Old Kent Bank are interested in comparing the reported value of customer savings account balances with their own findings regarding the actual value of such assets. Rather than reviewing the records of each savings account at the bank, the auditors decide to examine a representative sample of savings account balances. The frame from which they will sample is shown below.$75.30 $614.11 $696.34 $572.08$748.23 $21.20 $99.79 $1,233.38$530.40 $378.37 $596.14 $239.65$2,995.38 $1,069.06 $929.80 $259.98$123.65 $68.92 $192.35 $754.45$309.00 $163.31 $71.75 $904.92$40.70 $161.12 $459.38 $171.48$402.81 $157.44 $41.81 $87.08$489.97 $468.12 $400.57 $319.40$533.82 $1,801.35 $1,666.50 $37.16$85.92 $91.43 $193.14 $106.95$214.62 $10.62 $582.18 $39.65$123.66 $76.33 $291.73 $398.48$659.18 $101.24 $1,740.47 $322.26$1,509.34 $1,599.04 $358.62 $492.05$1,052.68 $596.33 $100.54 $1,288.70$421.46 $1,799.51 $581.21 $571.63$180.58 $98.82 $358.68 $38.93$874.78 $2,761.93 $750.44 $376.60$269.48 $456.79 $216.81 $305.49NARREND(A) What sample size would be required for the auditors to be approximately 95% sure that their estimate of the average savings account balance at this bank is within $150 of the true mean? Assume that their best estimate of the population standard deviation is $300.(B) Choose a simple random sample of the size found in (A).(C) Compute the observed sampling error based on the sample you have drawn from the population. How does the actual sampling error compare to the maximum possible probable absolute error established in (A)? Explain

Q: NARRBEGIN: SA_81_82A battery manufacturer wants to estimate the average number of defective (or dead) batteries contained in a box shipped by the company. Production personnel at this company have recorded the number of defective batteries found in each of the 2000 boxes shipped in the past week.NARREND (A) What sample size would be required for the production personnel to be approximately 95% sure that their estimate of the average number of defective batteries per box is within 0.3 unit of the true mean? Assume that the best estimate of the population standard deviation () is 0.9 defective batteries per box.(B) How does your answer to (A) change if the production personnel want their estimate to be within 0.5 unit of the actual population mean? Evaluate the tradeoff between required accuracy and sample size requirement for this case and the case in (A).

Q: A sales manager for a company that makes commercial ovens for restaurants is interested in estimating the average number of restaurants in all metropolitan areas across the entire country. He does not have access to the data for each metropolitan location, so he had decided to select a sample that will be representative of all such areas, and will use a sample size of 30. Do you believe that simple random sampling is the best approach to obtaining a representative subset of the metropolitan areas in the given frame? Explain. If not, recommend how the sales manager might proceed to select a better sample of size 30 from this data?

Q: Consider the frame of 50 full-time employees of Computer Technologies, Inc (CTI). CTI's human resources manager has collected annual salary figures for all employees and she has calculated a mean of $47,723, a median of $41,082 and a standard deviation of $24,167. A simple random sample of 10 employees is presented below (salary is in $1,000's). Compute the mean, median, and standard deviation for the sample and compare these statistics with the measures for the entire company.Employee12345678910Salary38.846.761.149.658.578.836.746.547.656.7

Q: The averaging effect says that as you average more and more observations from a given distribution, the variance of the average increases.

Q: The size of a sample can be selected by first determining the desired standard error and then using the formula to calculate n.

Q: The Central Limit Theorem (CLT) says that as long as the sample size is reasonably large, there is about a 95% chance that the magnitude of the sampling error for the mean will be no more than two standard errors.

Q: If the sample size is greater than 30, the Central Limit Theorem (CLT) will always apply.

Q: The Central Limit Theorem (CLT) states that the sampling distribution of the mean is approximately normal, no matter what the distribution of the population, so long as the sample size is large enough.

Q: Voluntary response bias occurs when the responses to questions do not reflect what the investigator had in mind.

Q: The randomized response technique is a way of getting at sensitive information to avoid estimation errors due to nontruthful responses.