A government agency reported that in 2020, 5.0% of people with a bachelor's degree and 8.5% of people with a high
school diploma but no college were unemployed. Suppose that the reported percentages were based on independently
selected representative samples of 600 people with a bachelor's degree and 600 people with a high school degree but no college.
It was also reported that 4.0% of those with master's degrees were unemployed in 2020. Suppose that this reported
percentage was based on a representative sample of 250 people with master's degrees. Use this information to answer
the following questions.
Construct a 99% large-sample confidence interval for the difference in the proportion who were unemployed in
2020 for those who have a bachelor's degree and those who have a master's degree. (Use
Bachelor's Degree − Master's Degree. Enter your answer using interval notation. Round your numerical values to
four decimal places.)
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To construct a confidence interval for the difference in proportions, we can use the formula:
CI = (p1 - p2) +/- z * sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2))
Where:
p1 = proportion of unemployed with a bachelor's degree (0.05)
p2 = proportion of unemployed with a master's degree (0.04)
n1 = sample size for bachelor's degree (600)
n2 = sample size for master's degree (250)
z = z-score for the desired confidence level (99%)
First, let's calculate the value inside the square root:
sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2))
= sqrt((0.05 * (1-0.05)/600) + (0.04 * (1-0.04)/250))
= sqrt((0.0475/600) + (0.0384/250))
= sqrt(0.0000791667 + 0.0001536)
= sqrt(0.0002327667)
≈ 0.015253
Next, let's calculate the z-score for the 99% confidence level. Since the confidence level is two-tailed, we need to find the z-score that corresponds to a cumulative probability of (1 - 0.01/2) = 0.995.
Using a standard normal distribution table or calculator, we find that the z-score is approximately 2.576.
Now we can calculate the confidence interval:
CI = (p1 - p2) +/- z * sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2))
= (0.05 - 0.04) +/- 2.576 * 0.015253
= (0.01) +/- 0.039287
Rounding to four decimal places, the 99% confidence interval for the difference in the proportion of unemployed between those with a bachelor's degree and those with a master's degree is (0.0100 - 0.0393, 0.0100 + 0.0393), or (-0.0293, 0.0493).
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