Chapter 9, Problem 66CE

a.

To determine

Construct a 95% confidence interval for the proportion favoring the Republican candidate.

Answer to Problem 66CE

The 95% confidence interval for the proportion favoring the Republican candidate is (0.489, 0.551).

Explanation of Solution

Calculation:

In this case, the number of voters favoring the Republican candidate is 520 $\left(=0.52\times 1,000\right)$.

Step-by-step procedure to find the 95% confidence interval for the proportion favoring the Republican candidate using MINITAB software:

• Choose Stat > Basic Statistics > 1 Proportion.
• Choose Summarized data.
• In Number of events, enter 520 and in Number of trials, enter 1,000.
• Check Options, enter Confidence level as 95.0.
• Choose not equal in alternative.
• Click OK in each dialog box.

Output is obtained as follows:

From the output, the 95% confidence interval for the proportion favoring the Republican candidate is (0.489, 0.551).

b.

To determine

Find the probability that the Democratic candidate is actually leading with.

Answer to Problem 66CE

The probability that the Democratic candidate is actually leading with is 0.102.

Explanation of Solution

Calculation:

In this case, the sample size (=1,000) is larger. Therefore, the mean and standard deviation of the sampling distribution using central limit theorem is $\left(p,\sqrt{\frac{p\left(1-p\right)}{n}}\right)$.

The mean (p) is 0.52 and the standard deviation is 0.0158 $\left(=\sqrt{\frac{\left(0.52\right)\times \left(1-0.52\right)}{1,000}}\right)$.

Hence, the sampling distribution follows normal with mean of 0.52 and standard deviation of 0.0158.

The probability that the Democratic candidate is actually leading is obtained as follows:

$\begin{array}{c}P\left(x<0.5\right)=\left(\frac{x-p}{\sqrt{\frac{p\left(1-p\right)}{n}}}<\frac{0.5-0.52}{0.0158}\right)\\ =P\left(z<-1.27\right)\end{array}$

Step-by-step procedure to find the probability value using MINITAB software:

• Choose Graph > Probability Distribution Plot >View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0 and Standard deviation as 1.
• Click the Shaded Area tab.
• Choose X Value and Left Tail for the region of the curve to shade.
• Enter the data value as –1.27.
• Click OK.

Output using MINITAB software is obtained as follows:

Thus, the probability that the Democratic candidate is actually leading is 0.102.

c.

To determine

Construct a 95% confidence interval for the proportion favoring the Republican candidate when the number of voters is 3,000.

Find the probability that the Democratic candidate is actually leading.

Answer to Problem 66CE

The 95% confidence interval for the proportion favoring the Republican candidate is (0.502, 0.538).

The probability that the Democratic candidate is actually leading is 0.0132.

Explanation of Solution

Calculation:

In this case, the number of voters favoring the Republican candidate is 1,560 $\left(=0.52\times 3,000\right)$.

Step-by-step procedure to find the 95% confidence interval for the proportion favoring the Republican candidate using MINITAB software:

• Choose Stat > Basic Statistics > 1 Proportion.
• Choose Summarized data.
• In Number of events, enter 1,560 and in Number of trials, enter 3,000.
• Check Options, enter Confidence level as 95.0.
• Choose not equal in alternative.
• Click OK in each dialog box.

Output is obtained as follows:

From the output, the 95% confidence interval for the proportion favoring the Republican candidate is (0.502, 0.538).

In this case, the sample size (=3,000) is larger. Therefore, the mean and standard deviation of the sampling distribution using central limit theorem is $\left(p,\sqrt{\frac{p\left(1-p\right)}{n}}\right)$.

The mean (p) is 0.52 and the standard deviation is 0.0091 $\left(=\sqrt{\frac{\left(0.52\right)\times \left(1-0.52\right)}{3,000}}\right)$.

Hence, the sampling distribution follows normal with mean of 0.52 and standard deviation of 0.0091.

The probability that the Democratic candidate is actually leading is obtained as follows:

$\begin{array}{c}P\left(x<0.5\right)=\left(\frac{x-p}{\sqrt{\frac{p\left(1-p\right)}{n}}}<\frac{0.5-0.52}{0.0091}\right)\\ =P\left(z<-2.20\right)\end{array}$

Step-by-step procedure to find the probability value using MINITAB software:

• Choose Graph > Probability Distribution Plot >View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0 and Standard deviation as 1.
• Click the Shaded Area tab.
• Choose X Value and Left Tail for the region of the curve to shade.
• Enter the data value as –2.20.
• Click OK.

Output using MINITAB software is obtained as follows:

Thus, the probability that the Democratic candidate is actually leading is 0.0139.