Question

Near the time of an election, a cable news service performs an opinion poll of 1,000 probable voters. It shows that the Republican contender has an advantage of 52% to 48%. Develop a 95% confidence interval for the proportion favoring the Republican candidate.

Answer 1

To develop a 95% confidence interval for the proportion favoring the Republican candidate, we can use the lower bound formula = p - Z * sqrt((p*(1-p))/n) and the upper bound formula = p + Z * sqrt((p*(1-p))/n). Plugging in the given values, the 95% confidence interval is approximately 0.495 to 0.545.

Step-by-step explanation:

To develop a 95% confidence interval for the proportion favoring the Republican candidate, we can use the formula:

Lower bound = p - Z * sqrt((p*(1-p))/n)

Upper bound = p + Z * sqrt((p*(1-p))/n)

where p is the observed proportion, Z is the z-score corresponding to the desired confidence level, and n is the sample size.

In this case, the observed proportion is 0.52, the sample size is 1000, and the desired confidence level is 95%.

Using a standard normal distribution table, the z-score for a 95% confidence level is approximately 1.96.

Plugging in these values, we can calculate:

Lower bound = 0.52 - 1.96 * sqrt((0.52*(1-0.52))/1000)

Upper bound = 0.52 + 1.96 * sqrt((0.52*(1-0.52))/1000)

Simplifying the calculations gives:

Lower bound ≈ 0.495

Upper bound ≈ 0.545

Therefore, the 95% confidence interval for the proportion favoring the Republican candidate is approximately 0.495 to 0.545.

Answer 2

To calculate a 95% confidence interval for the proportion favoring the Republican candidate, we use the sample proportion of 0.52, a z-score of 1.96, and the sample size of 1,000. The interval is computed to be from 48.9% to 55.1%, meaning there is a 95% confidence that the true proportion falls within this range.

Step-by-step explanation:

To develop a 95% confidence interval for the proportion favoring the Republican candidate, we use the formula for a confidence interval for a proportion, which is given by № + z*sqrt[№(1-№)/n], where № is the sample proportion, z is the z-score corresponding to the desired level of confidence, and n is the sample size.

The sample proportion (№) favoring the Republican candidate is 0.52. For a 95% confidence level, the appropriate z-score is approximately 1.96. The sample size (n) is 1,000. Plugging these values into the formula:

CI = 0.52 ± 1.96 * sqrt[0.52(0.48)/1000]

CI = 0.52 ± 1.96 * sqrt[0.2496/1000]

CI = 0.52 ± 1.96 * sqrt[0.0002496]

CI = 0.52 ± 1.96 * 0.0158

CI = 0.52 ± 0.031

So the 95% confidence interval is from 0.489 (48.9%) to 0.551 (55.1%).

This means we can be 95% confident that the true proportion of probable voters favoring the Republican candidate is between 48.9% and 55.1%.