Question

February 12, 2009 marked the 200th anniversary of Charles Darwin's birth. To celebrate, Gallup, a national polling organization, surveyed 1,018 randomly selected American adults about their education level and their beliefs about the theory of evolution. In their sample, 325 of their respondents had some college education and 228 were college graduates. Among the 325 respondents with some college education, 133 said that they believed in the theory of evolution. Among the 228 respondents who were college graduates, 121 said that they believed in the theory of evolution. Construct a 90% confidence interval for the difference between the proportions of college graduates and individuals with some college who believe in the theory of evolution. Round your sample proportions and margin of error to three decimal places.

Answer 1

To construct a 90% confidence interval for the difference between the proportions of college graduates and individuals with some college who believe in the theory of evolution, you can use the formula: CI = (p1 - p2) ± Z * sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2)). Substituting the given values into the formula, the confidence interval is (0.064, 0.178).

Step-by-step explanation:

To construct a 90% confidence interval for the difference between the proportions of college graduates and individuals with some college who believe in the theory of evolution, we can use the formula:

CI = (p1 - p2) ± Z * sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))

where:

• p1 is the sample proportion of college graduates who believe in evolution
• p2 is the sample proportion of individuals with some college who believe in evolution
• n1 is the sample size of college graduates
• n2 is the sample size of individuals with some college
• Z is the z-score corresponding to a 90% confidence level. From the standard normal distribution table, the z-score is approximately 1.645.

Given that there were 228 college graduates surveyed with 121 believing in evolution, the sample proportion p1 is 121/228 = 0.531. And for individuals with some college, there were 325 surveyed with 133 believing in evolution, so p2 is 133/325 = 0.410.

Substituting these values into the formula:

CI = (0.531 - 0.410) ± 1.645 * sqrt((0.531 * (1 - 0.531) / 228) + (0.410 * (1 - 0.410) / 325))

Calculating the margin of error using the formula:

ME = Z * sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))

we find:

ME = 1.645 * sqrt((0.531 * (1 - 0.531) / 228) + (0.410 * (1 - 0.410) / 325)) = 0.057

Therefore, the 90% confidence interval for the difference between the proportions is (0.531 - 0.410) ± 0.057, which simplifies to 0.121 ± 0.057.

Rounding to three decimal places, the confidence interval is (0.121 ± 0.057) or (0.064, 0.178).

Answer 2

A 90% confidence interval for the difference between the proportions of college graduates and individuals with some college who believe in the theory of evolution is (-0.044, 0.136). This means we are 90% confident that the true difference in proportions falls within this range.

Step-by-step explanation:

Calculate proportions:

College graduates: 121 believers / 228 total = 0.5307

Some college: 133 believers / 325 total = 0.4092

Calculate standard errors:

College graduates: sqrt(0.5307 * (1-0.5307) / 228) ≈ 0.0314

Some college: sqrt(0.4092 * (1-0.4092) / 325) ≈ 0.0295

Calculate margin of error:

Z-score for 90% confidence interval = 1.645

Margin of error = Z * sqrt(SE_graduates^2 + SE_some_college^2)

Margin of error ≈ 1.645 * sqrt(0.0314^2 + 0.0295^2) ≈ 0.0899

Construct the confidence interval:

Lower bound: 0.5307 - 0.0899 ≈ 0.4408

Upper bound: 0.5307 + 0.0899 ≈ 0.6206

Round and interpret:

Lower bound: -0.044 (rounded)

Upper bound: 0.136 (rounded)

Therefore, we are 90% confident that the true difference in proportions between college graduates and individuals with some college who believe in evolution falls between -0.044 and 0.136. This means that college graduates may be slightly more likely to believe in evolution, but the difference could also be zero or slightly negative.