From a random sample of 400 registered voters in one city,...

From a random sample of 400 registered voters in one city, 320 indicated that they would vote in favor of a proposed policy in an upcoming election. a. Calculate the LCL for a $98 \%$ confidence interval estimate for the population proportion in favor of this policy. b. Calculate the width of a $90 \%$ confidence interval estimate for the population proportion in favor of this policy.

From a random sample of 400 registered voters in one city, 320 indicated that they would vote in favor of a proposed policy in an upcoming election.
a. Calculate the LCL for a $98 \%$ confidence interval estimate for the population proportion in favor of this policy.
b. Calculate the width of a $90 \%$ confidence interval estimate for the population proportion in favor of this policy.

Key Concepts

Confidence Interval

A confidence interval is a range of values, constructed from sample data, that is likely to contain the unknown population parameter. It is calculated using the sample statistic along with a margin of error, and is defined at a specific confidence level (such as 98% or 90%), which reflects the degree of certainty in the estimate.

Margin of Error

The margin of error is the term added to and subtracted from the sample statistic to form the confidence interval. It is computed by multiplying the critical value (from the standard normal distribution, when appropriate) by the standard error, and it quantifies the precision of the estimate; a smaller margin suggests a more precise estimation.

Critical Value (Z-score)

The critical value, often expressed as a Z-score in the context of estimating proportions, represents the cutoff point on the standard normal distribution corresponding to the desired confidence level. It determines how wide the confidence interval is by indicating how many standard errors to span from the sample estimate.

Standard Error of the Proportion

The standard error of the proportion measures the variability of the sample proportion and is calculated using the formula ?[p(1-p)/n], where p is the sample proportion and n is the sample size. This standard error is crucial for determining the width of the confidence interval, as it reflects the inherent uncertainty in the estimate.

Sampling Distribution of Proportions

The sampling distribution of proportions refers to the distribution of sample proportions that would be obtained if many samples of the same size were drawn from the population. For large enough samples, this distribution approximates a normal distribution, enabling the use of z-scores to develop confidence intervals around the estimated proportion.

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