Exam 22: Binomial Distribution and Normal Approximation

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

SCENARIO 9-11 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 9-11,if you select a sample of 20 compact microwave ovens and are willing to have a level of significance of 0.01,the probability of making a Type II error is _____ if the mean power consumption of all such microwave ovens is in fact 257.3 W.

The changes in the price of the least consumed commodities are overly influential in a weighted aggregate price index.

If there is a small difference between the population parameter and the hypothesized value,the power of the test will be lower.

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 is the p-value of the test statistic using a 5% level of significance?

To test whether one proportion is higher than the other in two related samples,you can use the McNemar test.

In measuring the effect size of the difference between two population means,the rule of thumb suggests that a standardized effect size of around 0.2 can be classified as small.

The Paasche price index uses the consumption quantities in the year of interest as the weights.

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 more than 3 hours?

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 decision is to reject the null hypothesis when using a 5% level of significance.

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 exactly 2 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 at least 25 computers that require repair on a given day using a normal approximation.

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 0.01,the probability of the operations manager stopping the process if the population mean bulb life is 490 hours is .

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 are the simple price indices for wheat,corn,soybeans and milk,respectively,in 2008 using 2012 as the base year?

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 0.10,the power of the test is _____ if the mean power consumption of all such microwave ovens is in fact 257.3 W.

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 decision is to reject the null hypothesis when using a 10% level of significance.

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 large.

A simple price index tracks the price of a group of commodities at a given period of time to the price paid for that group of commodities at a particular point of time in the past.

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 market price of the houses in the suburban county constructed by the consulting group?

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 more than 0.2% of the items will be defective if the new machine is indeed functioning properly is .