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Chapter 6: Problem 13

Use the normal distribution to find a confidence interval for a proportion \(p\) given the relevant sample results. Give the best point estimate for \(p,\) the margin of error, and the confidence interval. Assume the results come from a random sample. A \(99 \%\) confidence interval for the proportion who will answer "Yes" to a question, given that 62 answered yes in a random sample of 90 people

Short Answer

Expert verified
The best point estimate for \( p \) is 0.6889, the margin of error is 0.0967, and the 99% confidence interval is (0.5922, 0.7856).

Step by step solution

01

Calculate the Point Estimate

To calculate the point estimate \( p̂ \), use the formula \( p̂ = x / n \) where x is the number of 'successes' (those who answered 'yes') and n is the total number in the sample. In this case, x=62 and n=90. So, \( p̂ = 62 / 90 = 0.6889 \).
02

Find the Z-Score

The problem specifies a 99% confidence interval. The Z-score corresponding with a 99% confidence interval is 2.57 (given or found in a standard Z-score table).
03

Calculate the Margin of Error

Next, find the margin of error using the formula \( E = Z * √( p̂ * (1 - p̂) / n ) \). Substituting the given values: \( E = 2.57 * √( 0.6889 * (1 - 0.6889) / 90 ) = 0.0967 \).
04

Find the Confidence Interval

Finally, find the 99% confidence interval using the formula \( CI = p̂ ± E \). Substituting the values calculated so far, \( CI = 0.6889 ± 0.0967 \), which simplifies to the interval (0.5922, 0.7856). Thus, with 99% confidence, the proportion of people who will answer 'Yes' lies between 0.5922 and 0.7856.

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Most popular questions from this chapter

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