Question

Greenville County, South Carolina, has 396,183 adult residents, of which 80,987 are 65 years or older. A survey wants to contact n = 689 residents. If repeated simple random samples of 689 residents are taken, what would be the range of the sample proportion of adults over 65 in the sample according to the 95 part of the 68-95-99.7 rule?

a. Lower bound: 0.183, Upper bound: 0.231

b. Lower bound: 0.195, Upper bound: 0.220

c. Lower bound: 0.172, Upper bound: 0.239

d. Lower bound: 0.185, Upper bound: 0.227

Answer 1

To find the range of the sample proportion for adults over 65 in random samples of 689 residents according to the 95% confidence level, one must calculate the population proportion, determine the standard error, and apply the z-score to find the margin of error. The lower and upper bounds of the confidence interval are then obtained by subtracting and adding the margin of error from the population proportion.

Step-by-step explanation:

The subject of the question is determining the range of the sample proportion for adults over 65 in repeated simple random samples of 689 residents, according to the 95% part of the 68-95-99.7 rule, also known as the Empirical Rule. In Greenville County, South Carolina, out of a total of 396,183 adult residents, 80,987 are aged 65 or older. To determine the range of the sample proportion, we calculate the proportion of adults over 65 in the population (p) and the standard error (SE) for the sample size (n).

To calculate the range:

1. Find the population proportion (p) by dividing the number of adults over 65 by the total number of adult residents: p = 80,987 / 396,183.
2. Calculate the standard error (SE) using the formula SE = sqrt(p(1-p)/n).
3. Determine the margin of error (ME) for the 95% confidence interval using the Z-score for 95% confidence, which is approximately 1.96. ME = Z * SE.
4. Calculate the lower and upper bounds of the 95% confidence interval: Lower bound = p - ME, Upper bound = p + ME.

After performing these calculations, the correct answer will be obtained.

Answer 2

To calculate the range of the sample proportion of adults over 65 in the sample, we can use the formula: Range = z * sqrt((p * (1-p)) / n). In this case, n = 689, p = 0.204, and z = 1.96. Plugging these values into the formula gives a range of 0.024.

Step-by-step explanation:

To calculate the range of the sample proportion of adults over 65 in the sample, we can use the formula:

Range = z * sqrt((p * (1-p)) / n)

In this case, n = 689, p = 80987/396183 = 0.204, and z can be calculated using the 68-95-99.7 rule. For a 95% confidence level, the z-value is approximately 1.96.

Plugging these values into the formula, we get:

Range = 1.96 * sqrt((0.204 * (1-0.204)) / 689) = 0.024

Therefore, the range of the sample proportion of adults over 65 in the sample is 0.024.