# Sample proportion question

When an election for political office takes place, the television networks cancel regular programming and instead provide election coverage. When the ballots are counted, the results are reported. However, for important offices such as president or senator in large states, the networks actively compete to see which will be the first to predict a winner. This is done through exit polls, wherein a random sample of voters who exit the polling booth is asked for whom they voted. From the data, the sample proportion of voters supporting the candidates is computed. Hypothesis testing is applied to determine whether there is enough evidence to infer that the leading candidate will garner enough votes to win. Suppose that in the exit poll from the state of Florida during the 2000 year elections, the pollsters recorded only the votes of the two candidates who had any chance of winning, Democrat Al Gore and Republican George W. Bush. In a sample of 765 voters, the number of votes cast for Al Gore was 358 and the number of votes cast for George W. Bush was 407. The network predicts the candidate as a winner if he wins more than 50% of the votes. The polls close at 8:00 P.M. Based on the sample results, should the networks announce at 8:01 P.M. that the Republican candidate George W. Bush will win the state? Select a level of significance by analyzing Type I and Type II errors and clearly show your analysis.

© BrainMass Inc. brainmass.com October 17, 2018, 10:10 am ad1c9bdddfhttps://brainmass.com/statistics/hypothesis-testing/sample-proportion-question-496443

#### Solution Preview

Hi there,

Here is my explanation:

The null hypothesis: P<=0.50

The alternative hypothesis: P>0.50

This is one tailed z test. At 0.05 significance level (type I error) the critical value ...

#### Solution Summary

The solution provides detailed explanation as to proportion based t test, type I error and type II error.

Questions about Hypothesis Testing of Mean & Proportion

Please provide explanations in detailed and attach all work on a words doc for explanations. Do all work on excel.

1. You are the manager of a restaurant that delivers pizza to college dormitory rooms. You have just changed your delivery in an effort to reduce the mean time between the order and completion of delivery from the current 25 minutes. A sample of 36 orders using the new delivery process yields a sample mean of 22.4 minutes and a sample standard deviation of 6 minutes.

a. Using the six-step critical value approach, at the 0.05 level of significance, is there evidence that the population mean delivery time has been reduced below the previous population mean value of 25 minutes?

b. At the 0.05 level of significance, use the five-step p-value approach.

c. Interpret the meaning of the p-value in (b)

d. Compare your conclusions in (a) and (b)

2. The U.S. Department of Education reports that 46% of full time college students are employed while attending college. A recent survey of 60 full-time students at Miami University found that 29 were employed.

a. Use the five-step p-value approach to hypothesis testing and a 0.05 level of significance to determine whether the proportion of full-time students at Miami University is different than the national norm of 0.46.

b. Assume that the study found that 36 of the 60 full-time students were employed and repeat (a). Are the conclusions the same?

3. One of the issues facing organizations is increasing diversity throughout the organization. One of the ways to evaluate an organization's success at increasing diversity is to compare the percentage of employees in the organization in a particular position with a specific background in the general workforce. Recently, a large academic medical center determined that 9 of 17 employees in a particular position were female, whereas 55% of the employees for this position in the general workforce were female. At the 0.05 level of significance, is there evidence that the proportion of females in this position at this medical center is different from what would be expected in the general workforce?

4. One of the biggest issues facing e-retailers is the ability to reduce the proportion of customers who cancel their transactions after they have selected their products. It has been estimated that about half of prospective customers cancel their transactions after they have selected their products. Suppose that a company changed its Web site so that customers could use a single page checkout system. Of these 500 customers, 210 cancelled their transactions after they had selected their products.

a. At the 0.01 level of significance, is there evidence that the population proportion of customers who select products and then cancel their transaction is less than 0.50 with the new system?

b. Suppose that a sample of n= 100 customers instead of n= 500 were provided with the new checkout system and that 42 of those customers cancelled their transactions after they had selected their products. At the 0.01 level of significance, is there evidence that the population proportion of customers who select products and then cancel their transaction is less than 0.50 with the new system?

c. Compare the results of (a) and (b) and discuss the effect that sample size has on the outcome, and, in general, in hypothesis testing.

5. An auditor for a government agency is assigned the task of evaluating reimbursement for office visits to physicians paid by Medicare. The audit was conducted on a sample of 75 of the reimbursements, with the following results:

- In 12 of the office visits, an incorrect amount of reimbursements was provided.

- The amount of reimbursement had a sample mean of $93.70 and a sample standard deviation of $34.55.

a. At the 0.05 level of significance, is there evidence that the population mean reimbursement was less than $100?

b. At the 0.05 level of significance, is there evidence that the proportion of incorrect reimbursements in the population was greater than 0.10?

c. Discuss the underlying assumptions of the test used in (a)

d. What is your answer to (a) if the sample mean equals $90?

e. What is your answer to (b) if office visits had incorrect reimbursements?

6. A manufacturing company produces electrical insulators. If the insulators break when in use, a short circuit is likely to occur. To test the strength of the insulators, destructive testing is carried out to determine how much force is required to break the insulators. Force is measured by observing the number of pounds of force applied to the insulator before it breaks. The following data are from 30 insulators subjected to this testing:

1870 1728 1656 1610 1634 1784 1522 1696 1592 1662

1866 1764 1734 1662 1734 1774 1550 1756 1762 1866

1820 1744 1788 1688 1810 1752 1680 1810 1652 1736

a. At the 0.05 level of significance, is there evidence that the population mean force is greater than 1,500 pounds?

b. What assumption about the population distribution is needed in order to conduct the t test in (a)?

c. Construct a histogram, boxplot , or normal probability plot to evaluate the assumption made in (b)

d. Do you think that the assumption needed in order to conduct the t test in (a) is valid? Explain.