The Residential Energy Consumption Survey found in 2001 that...

The Residential Energy Consumption Survey found in 2001 that $47 \%$ of American households had internet access. ${ }^{10}$ A market survey organization repeated this study in a certain town with 25,000 households, using a simple random sample of 500 households: 239 of the sample households had internet access. (a) The percentage of households in the town with internet access is estimated as (b) If possible, find a $95 \%$ -confidence interval for the percentage of all 25,000 households with internet access. If this is not possible, explain why not.

The Residential Energy Consumption Survey found in 2001 that $47 \%$ of American households had internet access. ${ }^{10}$ A market survey organization repeated this study in a certain town with 25,000 households, using a simple random sample of 500 households: 239 of the sample households had internet access.
(a) The percentage of households in the town with internet access is estimated as
(b) If possible, find a $95 \%$ -confidence interval for the percentage of all 25,000 households with internet access. If this is not possible, explain why not.

Key Concepts

Normal Approximation for Proportions

When constructing a confidence interval for a population proportion, the normal approximation to the binomial distribution is often used, provided that the sample size is large enough. This is typically justified if both the product of the sample size and the sample proportion and the product of the sample size and one minus the sample proportion are sufficiently large (commonly at least 10). This condition ensures the sampling distribution of the proportion is approximately normal.

Standard Error in Proportion Estimation

The standard error of a sample proportion quantifies the variability of the estimator from sample to sample. It is computed using the formula that involves the sample proportion and the sample size. This measure is crucial for constructing the confidence interval, as it determines how wide the interval will be.

Sample Proportion

In statistical inference, the sample proportion is used as a point estimate of the population proportion. It is calculated by dividing the number of successes observed in the sample by the total sample size. This measure provides an immediate estimate of the phenomenon under study within the population, assuming the sample is representative.

Confidence Interval for a Population Proportion

A confidence interval offers a range of values that, with a specified level of confidence (such as 95%), is believed to contain the true population proportion. It is constructed by taking the point estimate from the sample and adding and subtracting a margin of error, which is computed using the standard error and a critical value from the standard normal distribution.

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