Case Study – Election Results

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 the leading candidate will garner enough votes to win.

Suppose 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, conduct a one-sample hypothesis test to determine if the networks should announce at 8:01 P.M. the Republican candidate George W. Bush will win the state. Use 0.10 as the significance level (α).

Case 1: Election Results

•Use 0.10 as the significance level (α).

•Conduct a one-sample hypothesis test to determine if the networks should announce at 8:01 P.M. the Republican candidate George W. Bush will win the state.

In this case we want to test whether the data is giving enough evidence at 0.10 significance level to conclude that that percentage of voters who voted for George W. Bush is more than 50%. Thus the hypotheses we want to test are,

H0: p ≤ 0.50 and H1: p > 0.50

Now this is a right tailed, one sample test for proportion thus a Z test for proportion is the most appropriate test to use. Considering a 0.10 significance level the rejection region is,

Z test statistic> Critical value = Z0.10 = 1.28

From the given data,

Z test statistic = = 1.7716

As we can see that the test statistic is falling in the rejection region thus the null hypothesis should be rejected at 0.10 significance level. The conclusion based on the result is that the data is giving enough evidence, so the network should announce at 8:01 P.M. that the Republican candidate George W. Bush will win the state.

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Case Study – SpeedX

SpeedX, a large courier company, sends invoices to customers requesting payment within 30 days. The bill lists an address, and customers are expected to use their own envelopes to return their payments. Currently, the mean and standard deviation of the amount of time taken to pay bills are 24 days and 6 days, respectively. The chief financial officer (CFO) believes including a stamped self-addressed envelope would decrease the amount of time. She calculates the improved cash flow from a 2-day decrease in the payment period would pay for the costs of the envelopes and stamps. You have an MBA from the University of Phoenix, and work for SpeedX as a business analyst. One of your job duties is to run analytics and present the results to the senior management for critical decision-making. You see this as an opportunity to utilize some of the skills you gained in the Statistics course. Because of your strong understanding and background in inferential statistics, you decide to take up this important assignment. You have learned any analysis in inferential statistics starts with sampling. To test the CFO’s belief, you decide to randomly select 220 customers and propose to include a stamped self-addressed envelope with their invoices. The CFO accepts your proposal and allows you to run a pilot study. You then record the numbers of days until payment is received. Using your statistical expertise and skills you gained in the class, conduct a one-sample hypothesis test and determine if you can convince the CFO to conclude that the plan will be profitable. Use 0.10 and the significance level (α).

•Use 0.10 and the significance level (α).

•Conduct a one-sample hypothesis test and determine if you can convince the CFO to conclude the plan will be profitable.

As we can see, the plan would be profitable if the average numbers of days until payment is received decreases by more than 2 days. Hence the hypotheses which needed to be tested are,

H0: µ ≥ 22 and H1: µ < 22

We can see, this is a left tailed test for single population mean. As the population standard deviation is known so a Z-test for mean is the most appropriate test.

Considering a 0.10 significance level the rejection region is,

Z test statistic Critical value = -Z0.10 = -1.28

From the given data,

Z test statistic = = -0.9102

As we can see that the test statistic is not falling in the rejection region thus the null hypothesis should not be rejected at 0.10 significance level. The conclusion based on the result is that the data is not giving enough evidence that the plan would be profitable.

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