Question

Early in a presidential primary, a survey was conducted in which a random sample of 40 voters was asked which candidate, A through E, they would likely support for the nomination for president. The results are provided in the accompanying table. The letter N indicates that the voter had no opinion.

a. Determine the sample proportion who support candidate A.
b. Calculate a 95% confidence interval for the population proportion of voters who support candidate A.
c. Determine the sample proportion who have no opinion.
d. Calculate a 99% confidence interval for the population proportion of voters who have no opinion.

Answer 1

a. The sample proportion who support candidate A is 0.45. b. The 95% confidence interval for the population proportion of voters who support candidate A is (0.2701, 0.6299). c. The sample proportion who have no opinion is 0.125. d. The 99% confidence interval for the population proportion of voters who have no opinion is (-0.0596, 0.3096).

Step-by-step explanation:

a. The sample proportion who support candidate A can be determined by dividing the number of voters who support candidate A by the total number of voters in the sample. In this case, there are 18 voters who support candidate A out of a total of 40 voters, so the sample proportion is 18/40 = 0.45 or 45%.

b. To calculate a 95% confidence interval for the population proportion of voters who support candidate A, you can use the formula:

Confidence Interval = sample proportion ± (critical value) * (standard error)

The critical value for a 95% confidence level is approximately 1.96. The standard error can be calculated as:

Standard Error = sqrt((sample proportion * (1 - sample proportion)) / sample size)

Using the sample proportion from part a (0.45) and the sample size (40), we can calculate the standard error as: Standard Error = sqrt((0.45 * (1 - 0.45)) / 40) = 0.0918.

Plugging these values into the confidence interval formula, we get:

Confidence Interval = 0.45 ± (1.96 * 0.0918) = 0.45 ± 0.1799. Therefore, the 95% confidence interval for the population proportion of voters who support candidate A is (0.2701, 0.6299).

c. The sample proportion who have no opinion can be determined by dividing the number of voters with no opinion by the total number of voters in the sample. In this case, there are 5 voters with no opinion out of a total of 40 voters, so the sample proportion is 5/40 = 0.125 or 12.5%.

d. To calculate a 99% confidence interval for the population proportion of voters who have no opinion, you can follow the same steps as in part b, using the sample proportion from part c (0.125) and the sample size (40). The critical value for a 99% confidence level is approximately 2.58. The standard error can be calculated as: Standard Error = sqrt((0.125 * (1 - 0.125)) / 40) = 0.0714.

Plugging these values into the confidence interval formula, we get: Confidence Interval = 0.125 ± (2.58 * 0.0714) = 0.125 ± 0.1846. Therefore, the 99% confidence interval for the population proportion of voters who have no opinion is (-0.0596, 0.3096).