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Chapter 10: Problem 74

Sixty-five percent of all male voters and \(40 \%\) of all female voters favor a particular candidate. A sample of 100 male voters and another sample of 100 female voters will be polled. What is the probability that at least 10 more male voters than female voters will favor this candidate?

Short Answer

Expert verified
The probability is approximately 98.3%.

Step by step solution

01

Setting Up the Binomial Distributions

Let X and Y be the number of male and female voters who favor the candidate, respectively. Then X follows a binomial distribution with parameters n=100 and p=0.65, and Y follows a binomial distribution with parameters n=100 and p=0.40, symbolically written as X~B(100,0.65) and Y~B(100,0.40).
02

Calculating the Expected Difference

The expected difference between the number of male and female voters who favor the candidate is E[X-Y] = E[X] - E[Y] = np for X - np for Y = 100*0.65 - 100*0.40 = 25, which means we expect on average 25 more male voters than female voters to favor the candidate.
03

Calculating the Standard Deviation of the Difference

The standard deviation of the difference between the number of male and female voters who favor the candidate can be found using the formula sqrt(np(1-p)) for each group, and it is sqrt(100*0.65*0.35) = 5 for male voters and sqrt(100*0.40*0.60) = 5 for female voters. The variance of X-Y is the sum of the variances of X and Y, so the standard deviation of X-Y is sqrt(5^2+5^2) = 7.07.
04

Norm Approximation and Finding the Z-Score

Since both X and Y are binomial and n is large, the Central Limit Theorem applies, and we can approximate the distribution of X-Y by a normal distribution. We want to find P(X-Y >= 10), which is equivalent to P((X-Y-25)/7.07 >= (10-25)/7.07). This is a standard normal distribution, so the Z-score is z = (10-25)/7.07 = -2.12.
05

Calculating the Probability

The probability that the Z-score is more than -2.12 is 0.983, according to the standard normal distribution table. So there is approximately a 98.3% chance that the difference in the number of male and female voters favoring the candidate will be at least 10.

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Most popular questions from this chapter

Sample of 500 observations taken from the first population gave \(x_{1}=305\). Another sample of 600 observations taken from the second population gave \(x_{2}=348\). a. Find the point estimate of \(p_{1}-p_{2}\). b. Make a \(97 \%\) confidence interval for \(p_{1}-p_{2}\). c. Show the rejection and nonrejection regions on the sampling distribution of \(\hat{p}_{1}-\hat{p}_{2}\) for \(H_{0}: p_{1}=p_{2}\) versus \(H_{1}: p_{1}>p_{2}\). Use a significance level of \(2.5 \%\). d. Find the value of the test statistic \(z\) for the test of part \(\mathrm{c}\). e. Will you reject the null hypothesis mentioned in part \(\mathrm{c}\) at \(\mathrm{a}\) significance level of \(2.5 \% ?\)

The management at New Century Bank claims that the mean waiting time for all customers at its branches is less than that at the Public Bank, which is its main competitor. A business consulting firm took a sample of 200 customers from the New Century Bank and found that they waited an average of \(4.5\) minutes before being served. Another sample of 300 customers taken from the Public Bank showed that these customers waited an average of \(4.75\) minutes before being served. Assume that the standard deviations for the two populations are \(1.2\) and \(1.5\) minutes, respectively. a. Make a \(97 \%\) confidence interval for the difference between the two population means. b. Test at a \(2.5 \%\) significance level whether the claim of the management of the New Century Bank is true. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.01\) ? What if \(\alpha=.05\) ?

A town that recently started a single-stream recycling program provided 60 -gallon recycling bins to 25 randomly selected households and 75 -gallon recycling bins to 22 randomly selected households. The total volume of recycling over a 10-week period was measured for each of the households. The average total volumes were 382 and 415 gallons for the households with the 60 - and 75 -gallon bins, respectively. The sample standard deviations were \(52.5\) and \(43.8\) gallons, respectively. Assume that the 10 -week total volumes of recycling are approximately normally distributed for both groups with unknown and unequal population standard deviations. a. Construct a \(98 \%\) confidence interval for the difference in the mean volumes of 10-week recyclying for all households with the 60 - and 75 -gallon bins. b. Using a \(2 \%\) significance level, can you conclude that the average 10-week recycling volume of all households having 60-gallon containers is different from the average 10 -week recycling volume of all households that have 75 -gallon containers? c. Suppose that the sample standard deviations were \(59.3\) and \(33.8\) gallons, respectively. Redo parts a and b. Discuss any changes in the results.

Refer to the previous exercise. Suppose Gamma Corporation decides to test governors on seven cars. However, the management is afraid that the speed limit imposed by the governors will reduce the number of contacts the salespersons can make each day. Thus, both the fuel consumption and the number of contacts made are recorded for each car/salesperson for each week of the testing period, both before and after the installation of governors. Suppose that as a statistical analyst with the company, you are directed to prepare a brief report that includes statistical analysis andinterpretation of the data. Management will use your report to help decide whether or not to install governors on all salespersons' cars. Use \(90 \%\) confidence intervals and \(.05\) significance levels for any hypothesis tests to make suggestions. Assume that the differences in fuel consumption and the differences in the number of contacts are both normally distributed.

A sample of 1000 observations taken from the first population gave \(x_{1}=290\). Another sample of 1200 observations taken from the second population gave \(x_{2}=396\). a. Find the point estimate of \(p_{1}-p_{2}\). b. Make a \(98 \%\) confidence interval for \(p_{1}-p_{2}\). c. Show the rejection and nonrejection regions on the sampling distribution of \(\hat{p}_{1}-\hat{p}_{2}\) for \(H_{0}: p_{1}=p_{2}\) versus \(H_{1}: p_{1}
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