(a) According to a report in the Roanoke Times,...

(a) According to a report in the Roanoke Times, approximately $2 / 3$ of 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, approximately $2 / 3$ of 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

Critical Value

The critical value is a multiplier used in the construction of confidence intervals that corresponds to the desired level of confidence. For a 95% confidence interval, the critical value is typically derived from the standard normal distribution (often approximated as 1.96) when sample sizes are sufficiently large. It determines how many standard errors away from the sample estimate the interval extends.

Margin of Error

The margin of error reflects the maximum amount by which the estimated sample proportion is expected to differ from the true population proportion within a certain confidence level. It is calculated by multiplying the critical value associated with the chosen confidence level by the standard error. The margin of error provides insight into the precision of the estimate.

Standard Error of a Proportion

The standard error of a proportion is a measure of the variability of the sampling distribution of the sample proportion. It is calculated using the formula ?[p(1 - p) / n], where p is the sample proportion and n is the sample size. This value gives an idea of how much the sample proportion is expected to vary from the true proportion from one sample to another.

Confidence Interval for a Proportion

A confidence interval for a proportion is a range of values, derived from sample data, that is likely to contain the true population proportion. It is constructed by adding and subtracting a margin of error from the sample proportion, giving a probabilistic guarantee (such as 95% confidence) that this interval includes the population parameter.

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