Exam 22: Binomial Distribution and Normal Approximation

If the number of blocks in the experiment is larger than 5,the Friedman rank test statistic can be approximated by a standardized normal distribution.

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 8-16,95% of the stores have sold between 5.623 and 6.377 lawnmowers.

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 7-7,80% of all the samples of 4 will have a sample mean of no more than how many hours from the population mean?

SCENARIO 16-15 Given below are the average prices for three types of energy products for five consecutive years. -Referring to Scenario 16-15,what is the Paasche price index for the group of three energy items in year 5 for a family that consumed 13 units of electricity,26 units of natural gas and 235 units of fuel oil in year 5 using year 1 as the base year?

For a price index,it is preferable to select the base period in a period of economic stability.

SCENARIO 9-15 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 9-15,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 marketing the new local anesthetic when it is better is _____ if the population mean effective time is 8 minutes.

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 none of the items will be defective if the new machine is indeed functioning properly is .

As a general rule,one can use the normal distribution to approximate a binomial distribution whenever n $\pi$ and n( $\pi$ -1) are at least 5.

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 are the lower and upper critical values of the test?

Practical significance can eventually be found when the sample size becomes large enough.

In measuring the effect size of the difference between two population means,the rule of thumb suggests that a standardized effect size of around 0.8 can be classified as medium.

If the number of blocks in the experiment is larger than 5,the Friedman rank test statistic can be approximated by a chi-square distribution.

SCENARIO 8-13 The president of a university is concerned that illicit drug use on campus is higher than the 5% targeted level.A random sample of 250 students from a population of 2,000 revealed that 7 of them had used illicit drugs during the last 12 months. -Referring to Scenario 8-13,using the 90% one-sided confidence interval,the president can be 85% confident that no more than 5% of the students at the university had used illicit drugs during the last 12 months.

SCENARIO 9-15 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 9-15,if you select a sample of 25 new local anesthetics and are willing to have a level of significance of 0.1,the power of the test is _____ if the population mean effective time is 8 minutes.

SCENARIO 9-13 A manufacturer produces light bulbs that have a mean life of at least 500 hours when the production process is working properly.Based on past experience,the population standard deviation is 50 hours and the light bulb life is normally distributed.The operations manager stops the production process if there is evidence that the population mean light bulb life is below 500 hours. -Referring to Scenario 9-13,if you select a sample of 100 light bulbs and are willing to have a level of significance of 0.05,the probability of the operations manager stopping the process when the process is not working properly is _____ if the population mean bulb life is 490 hours.

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 differs from the population mean by less than 2 hours?

The McNemar test is approximately distributed as a Student's t.

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,what proportion of the samples will have less than 15% of customers who will make a purchase after visiting the web site?

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,which of the following is one of the properties required so that the binomial distribution can be used to compute the probability that no more than 2 computers will require repair on a given day?

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 no more than 0.2% of the items will be defective if the new machine is indeed functioning properly is .