8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

2:15 minutes

Problem 7.1.34b

Textbook Question

Astrology A sociologist plans to conduct a survey to estimate the percentage of adults who believe in astrology. How many people must be surveyed if we want a confidence level of 99% and a margin of error of four percentage points?

b. Use the information from a previous Harris survey in which 26% of respondents said that they believed in astrology.

Verified step by step guidance

Determine the formula for the required sample size when estimating a population proportion. The formula is:

n = \frac{(^{z)}}{2}

, where

z

is the z-score corresponding to the confidence level,

p

is the estimated proportion, and

E

is the margin of error.

Identify the values given in the problem: the confidence level is 99%, so the z-score corresponding to this level is approximately 2.576. The margin of error is 4%, or 0.04. From the previous survey, the estimated proportion

p

is 0.26.

Substitute the values into the formula:

n = \frac{(^{2.576)}}{2}

Simplify the expression step by step: First, calculate

1 - 0.26

, then multiply

0.26

by the result. Next, square the z-score

2.576

, and multiply it by the product of

0.26 (1 - 0.26)

. Finally, divide by the square of the margin of error

0.04

Round the result to the nearest whole number, as the sample size must be a whole number. This will give you the minimum number of people that must be surveyed to achieve the desired confidence level and margin of error.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the confidence interval. A 99% confidence level means that if the survey were repeated multiple times, 99% of the calculated intervals would contain the true percentage of adults who believe in astrology. This high level of confidence typically requires a larger sample size.

Margin of Error

The margin of error indicates the range within which the true population parameter is expected to fall, based on the sample results. In this case, a margin of error of four percentage points means that if the survey estimates that 26% of adults believe in astrology, the actual percentage is likely between 22% and 30%. A smaller margin of error requires a larger sample size.

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of respondents needed to achieve a desired level of confidence and margin of error. It involves using the estimated proportion (in this case, 26%) along with the desired confidence level and margin of error to ensure that the survey results are statistically valid and reliable.

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