Exam 22: Binomial Distribution and Normal Approximation

SCENARIO 8-14 The president of a university is concerned that the percentage of students who have cheated on an exam is higher than the 1% acceptable level.A confidential random sample of 1,000 students from a population of 7,000 revealed that 6 of them said that they had cheated on an exam during the last semester. -Referring to Scenario 8-14,using the 90% one-sided confidence interval,the president can be 95% confident that no more than 1% of the students at the university had cheated on an exam during the last semester.

Unweighted aggregate price indices account for differences in the magnitude of prices per unit and differences in the consumption levels of the items in the market basket.

SCENARIO 7-7 Times spent studying by students in the week before final exams follow a normal distribution with standard deviation 8 hours.A random sample of 4 students was taken from a population of 50 in order to estimate the mean study time for the population of all students.Use the finite population correction. -Referring to Scenario 7-7,what is the probability that the sample mean is more than 3 hours below the population mean?

SCENARIO 12-19 The director of the MBA program of a state university wanted to know if a one week orientation would change the proportion among potential incoming students who would perceive the program as being good.Given below is the result from 215 students' view of the program before and after the orientation. -Referring to Scenario 12-19,what should be the director's conclusion?

In measuring the effect size of the difference between two population means,the rule of thumb suggests that a standardized effect size of around 0.5 can be classified as medium.

SCENARIO 9-12 A drug company is considering marketing a new local anesthetic.The effective time of the anesthetic the drug company is currently producing has a normal distribution with a mean of 7.4 minutes with a standard deviation of 1.2 minutes.The chemistry of the new anesthetic is such that the effective time should be normally distributed with the same standard deviation.The company will market the new local anesthetic as being better if there is evidence that the population mean effective time is greater than the 7.4 minutes of the current local anesthetic. -Referring to Scenario 9-12,if you select a sample of 25 new local anesthetics and are willing to have a level of significance of 0.05,the probability of the company not marketing the new local anesthetic when it is not better is .

SCENARIO 12-18 The director of transportation of a large company is interested in the usage of the company's van pool program.She surveyed 129 of her employees on the usage of the program before and after a campaign to convince her employees to use the service and obtained the following: She will use this information to perform test using a level of significance of 0.05. -Referring to Scenario 12-18,the director now wants to know if the proportion of employees who use the service before the campaign and the proportion of employees who use the service after the campaign are the same.What is the value of the test statistic using a 5% level of significance?

SCENARIO 9-12 A drug company is considering marketing a new local anesthetic.The effective time of the anesthetic the drug company is currently producing has a normal distribution with a mean of 7.4 minutes with a standard deviation of 1.2 minutes.The chemistry of the new anesthetic is such that the effective time should be normally distributed with the same standard deviation.The company will market the new local anesthetic as being better if there is evidence that the population mean effective time is greater than the 7.4 minutes of the current local anesthetic. -Referring to Scenario 9-12,if you select a sample of 25 new local anesthetics and are willing to have a level of significance of 0.05,the probability of the company failing to market the new local anesthetic when it is better is _____ if the population mean effective time is 8 minutes.

SCENARIO 7-8 According to a survey,only 15% of customers who visited the web site of a major retail store made a purchase.Random samples of size 50 are selected from a population of 900.Use the finite population correction factor. -Referring to Scenario 7-8,the requirements for using a normal distribution to approximate a binomial distribution is fulfilled.

The use of the finite population correction factor when sampling without replacement from finite populations will

SCENARIO 12-21 A perfume manufacturer is trying to choose between 2 magazine advertising layouts.An expensive layout would include a small package of the perfume.A cheaper layout would include a "scratch- and-sniff" sample of the product.The manufacturer would use the more expensive layout only if there is evidence that it would lead to a higher approval rate.The manufacturer presents both layouts to 5 groups and determines the approval rating from each group on both layouts.The data are given below.Use this to test whether the median difference in approval rating is different from zero in favor of the more expensive layout with a level of significance of 0.05. -Referring to Scenario 12-21,what is the right test to use?

Based on past experience,only 0.5% of the invoices of a company contain an error.Out of the 1,500 invoices that the company will issue,what is the approximate probability that zero invoices will contain an error?

SCENARIO 6-7 A company has 125 personal computers.The probability that any one of them will require repair on a given day is 0.15. -Referring to Scenario 6-7 and assuming that the number of computers that requires repair on a given day follows a binomial distribution,compute the probability that there will be less than 25 or more than 30 computers that require repair on a given day using a normal approximation.

If a new machine of a production plant is functioning properly,only 1% of the items produced will be defective.Out of the 1,000 items the plant produces on a single day,the approximate probability that less than 0.2% of the items will be defective if the new machine is indeed functioning properly is .

Based on past experience,only 0.5% of the invoices of a company contain an error.Out of the 1,500 invoices that the company will issue,what is the approximate probability that more than 4 invoices will contain an error?

SCENARIO 8-16 A random sample of 100 stores from a large chain of 500 garden supply stores was selected to determine the mean number of lawnmowers sold at an end-of-season clearance sale.The sample results indicated a mean of 6 and a standard deviation of 2 lawnmowers sold.A 95% confidence interval (5.623 to 6.377)was established based on these results. -Referring to Scenario 8-16,the confidence interval estimate of the mean with finite population correction will be wider in range than confidence interval estimate without finite population correction.

SCENARIO 6-7 A company has 125 personal computers.The probability that any one of them will require repair on a given day is 0.15. -Referring to Scenario 6-7 and assuming that the number of computers that requires repair on a given day follows a binomial distribution,compute the probability that there will be no more than 8 computers that require repair on a given day using a normal approximation.

SCENARIO 7-7 Times spent studying by students in the week before final exams follow a normal distribution with standard deviation 8 hours.A random sample of 4 students was taken from a population of 50 in order to estimate the mean study time for the population of all students.Use the finite population correction. -Referring to Scenario 7-7,90% of all the samples of size 4 will have a sample mean of no more than how many hours from the population mean?

SCENARIO 9-14 An appliance manufacturer claims to have developed a compact microwave oven that consumes a population mean of no more than 250 W.From previous studies,it is believed that power consumption for microwave ovens is normally distributed with a population standard deviation of 15 W.If there is evidence that the population mean consumption is greater than 250 W,the manufacturer will be unable to make the claim. -Referring to Scenario 9-14,if you select a sample of 20 compact microwave ovens and are willing to have a level of significance of 0.01,the power of the test is _____ if the mean power consumption of all such microwave ovens is in fact 257.3 W.

The McNemar test is approximately distributed as a standardized normal random variable.