Exam 22: Binomial Distribution and Normal Approximation

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 incorrectly marketing the new local anesthetic when it is not better is .

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 Laspeyres price index for the group of three energy items in year 4 for a family that consumed 15 units of electricity,24 units of natural gas and 200 units of fuel oil in year 1 using year 1 as the base year?

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,what is the critical value for the 90% one-sided confidence interval for the proportion of students who had cheated on an exam during the last 12 months?

SCENARIO 12-20 A filling machine at a local soft drinks company is calibrated to fill the cans at a mean amount of 12 fluid ounces and a standard deviation of 0.5 ounces.The company wants to test whether the standard deviation of the amount filled by the machine is 0.5 ounces.A random sample of 15 cans filled by the machine reveals a standard deviation of 0.67 ounces. -Referring to Scenario 12-20,in order to perform the test,you need to assume that the amount filled by the machine follows a normal distribution.

SCENARIO 8-15 A wealthy real estate investor wants to decide whether it is a good investment to build a high-end shopping complex in a suburban county in Chicago.His main concern is the total market value of the 3,605 houses in the suburban county.He commissioned a statistical consulting group to take a sample of 200 houses and obtained a sample mean market price of $225,000 and a sample standard deviation of $38,700.The consulting group also found out that the mean differences between market prices and appraised prices was $125,000 with a standard deviation of $3,400.Also,the proportion of houses in the sample that are appraised for higher than the market prices is 0.24. -Referring to Scenario 8-15,what will be the 90% confidence interval for the total difference between the market prices and appraised prices of the houses in the suburban county constructed by the consulting group?

If we are performing a two-tail test of whether μ = 100,the probability of detecting a shift of the mean to 105 will be _____ the probability of detecting a shift of the mean to 110.

When you use the Poisson distribution to approximate the binomial distribution,the mean is n $\pi$ where n is the sample size and $\pi$ is the probability of an event of interest.

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 unweighted aggregate price index for the group of three energy items in year 3 using year 1 as the base year?

To determine the probability of getting between 3 and 4 events of interest in a binomial distribution,you will find the area under the normal curve between X = 3.5 and 4.5.

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 is the value of the test statistic?

You have 5 stocks in your investment portfolio.You want to keep track of your portfolio's performance over the last 2 years.You have data on each of the stock's average price and the number of shares of each stock for every day of the last 2 years.Which of the following would be the most appropriate analysis to perform?

SCENARIO 12-20 A filling machine at a local soft drinks company is calibrated to fill the cans at a mean amount of 12 fluid ounces and a standard deviation of 0.5 ounces.The company wants to test whether the standard deviation of the amount filled by the machine is 0.5 ounces.A random sample of 15 cans filled by the machine reveals a standard deviation of 0.67 ounces. -Referring to Scenario 12-20,the parameter of interest in the test is .

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,there are 10 possible samples of 100 stores that can be selected out of the population of 1,000 stores.

SCENARIO 16-16 Given below are the prices of a basket of four food items from 2008 to 2012. -Referring to Scenario 16-16,what is the Laspeyres price index for the basket of four food items in 2011 that consisted of 50 bushels of wheat,30 bushels of corn,40 bushels of soybeans and 80 hundredweight of milk in 2008 using 2008 as the base year?

One of the reasons that a correction for continuity adjustment is needed when approximating the binomial distribution with a normal distribution is because the normal distribution is used for a discrete random variable while the binomial distribution is used for a continuous random variable.

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 248 W.

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 superintendent can be 85% confident that no more than 1% of the students at the university had cheated on an exam during the last semester.

To construct a bootstrap confidence interval estimate for the population mean,you will first select a random sample of size n without replacement from a population of size N as the initial sample.

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 are the simple price indices for electricity,natural gas and fuel oil,respectively,in year 1 using year 5 as the base year?

As a general rule,one can use the normal distribution to approximate a binomial distribution whenever the sample size is at least 30.