Answer:
Explanation:
To find the value of Z for a 95% confidence level, we can use a standard normal distribution table or a calculator that has a built-in function for finding Z values.
Using a calculator, we can use the following steps:
Determine the level of confidence, which is 95%. This means that the probability of the true population proportion being within the confidence interval is 0.95.
Find the critical value of Z using a Z-table or calculator. For a 95% confidence level, the critical Z value is 1.96.
Calculate the sample proportion, which is the number of married couples in the sample with at least one partner having a doctorate degree divided by the total sample size:
p-hat = 41/320 = 0.128125
Calculate the standard error of the sample proportion, which is the square root of the product of the sample proportion and the complement of the sample proportion, divided by the sample size:
SE(p-hat) = sqrt((p-hat)(1 - p-hat)/n) = sqrt((0.128125)(1 - 0.128125)/320) = 0.0248 (rounded to four decimal places)
Calculate the margin of error, which is the product of the critical Z value and the standard error:
Margin of error = Z * SE(p-hat) = 1.96 * 0.0248 = 0.0486 (rounded to four decimal places)
Calculate the lower and upper bounds of the confidence interval by subtracting and adding the margin of error to the sample proportion:
Lower bound = p-hat - margin of error = 0.128125 - 0.0486 = 0.0795 (rounded to four decimal places)
Upper bound = p-hat + margin of error = 0.128125 + 0.0486 = 0.1767 (rounded to four decimal places)
Therefore, the 95% confidence interval for the percentage of married couples in which at least one partner has a doctorate degree is (0.0795, 0.1767).