(a) A random sample of 200 voters in a town is selected, and...

(a) A random sample of 200 voters in a town is selected, and 114 are found to support an annexation suit. Find the $96 \%$ confidence interval for the fraction of the voting population favoring the suit. (b) What can we assert with $96 \%$ confidence about the possible size of our error if we estimate the fraction of voters favoring the annexation suit to be $0.57 ?$

(a) A random sample of 200 voters in a town is selected, and 114 are found to support an annexation suit. Find the $96 \%$ confidence interval for the fraction of the voting population favoring the suit.
(b) What can we assert with $96 \%$ confidence about the possible size of our error if we estimate the fraction of voters favoring the annexation suit to be $0.57 ?$

Key Concepts

Standard Error of a Proportion

The standard error quantifies the variability in the sample proportion estimate and is calculated using the sample proportion and the sample size. It measures the dispersion of the sampling distribution, which is essential for constructing confidence intervals.

Critical Value (Z-score)

The critical value, typically derived from the standard normal distribution, is used to determine the number of standard errors to add and subtract from the sample estimate. It is chosen based on the desired level of confidence and is integral in calculating the margin of error.

Sample Proportion

The sample proportion is the fraction of the sampled individuals who exhibit a particular characteristic, serving as an estimate of the true population proportion. This estimate forms the basis for constructing confidence intervals in proportion-related studies.

Margin of Error

The margin of error represents the maximum amount by which the sample estimate is expected to differ from the true population parameter, given the chosen confidence level. It is calculated using the standard error of the estimate and the critical value corresponding to the desired level of confidence.

Confidence Interval for a Proportion

This concept refers to a range of values, derived from sample data, within which the true population proportion is estimated to lie with a certain level of confidence. The confidence interval accounts for sample variability and uncertainty, providing a probabilistic estimate of the parameter.

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