(a) According to a report in the Roanoke Times & World-News,...

(a) According to a report in the Roanoke Times \& World-News, approximately $2 / 3$ of the 1600 adults polled by telephone said they think the space shuttle program is a good investment for the country. Find a $95 \%$ confidence interval for the proportion of American adults who think the space shuttle program is a good investment for the country. (b) What can we assert with $95 \%$ confidence about the possible size of our error if we estimate the proportion of American adults who think the space shuttle program is a good investment to be $2 / 3 ?$

(a) According to a report in the Roanoke Times \& World-News, approximately $2 / 3$ of the 1600 adults polled by telephone said they think the space shuttle program is a good investment for the country. Find a $95 \%$ confidence interval for the proportion of American adults who think the space shuttle program is a good investment for the country.
(b) What can we assert with $95 \%$ confidence about the possible size of our error if we estimate the proportion of American adults who think the space shuttle program is a good investment to be $2 / 3 ?$

Key Concepts

Central Limit Theorem

The Central Limit Theorem is a foundational concept in statistics stating that, for a large enough sample size, the sampling distribution of the sample mean or proportion will approach a normal distribution, regardless of the population's distribution. This theorem enables statisticians to use normal probability models to make inference about population parameters even when the underlying distribution is not normal.

Sampling Distribution of the Proportion

The sampling distribution of the proportion refers to the distribution of sample proportions computed from many independent samples drawn from the same population. For sufficiently large sample sizes, thanks to the Central Limit Theorem, this distribution approximates a normal distribution, which justifies the use of z-scores in constructing confidence intervals.

Confidence Interval for a Proportion

A confidence interval for a proportion provides a range of values within which one can expect the true population proportion to fall, with a specified level of confidence (e.g., 95%). This interval is constructed around a sample proportion and takes into account the variability expected in repeated samples, allowing researchers to infer information about the whole population.

Margin of Error

The margin of error quantifies the extent of uncertainty associated with an estimate. It is calculated by multiplying a critical value (derived from the standard normal distribution for large samples) by the standard error of the proportion. This term specifies the maximum expected difference between the sample proportion and the true population proportion within the confidence interval.

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