In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.
(0.512, 0.596)

Verified step by step guidance

1. Identify the given confidence interval, which is (0.512, 0.596). The lower bound is 0.512, and the upper bound is 0.596.
2. To find the margin of error (E), use the formula: $E\; =\; \backslash frac\{\backslash text\{Upper\; Bound\}\; -\; \backslash text\{Lower\; Bound\}\}\{2\}$. Substitute the values of the upper and lower bounds into this formula.
3. To find the sample proportion (p̂), use the formula: $\backslash hat\{p\}\; =\; \backslash frac\{\backslash text\{Upper\; Bound\}\; +\; \backslash text\{Lower\; Bound\}\}\{2\}$. Substitute the values of the upper and lower bounds into this formula.
4. Perform the subtraction and division in the margin of error formula to calculate E.
5. Perform the addition and division in the sample proportion formula to calculate p̂.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed as an interval (e.g., (0.512, 0.596)) and is associated with a confidence level, typically 95% or 99%. This means that if we were to take many samples and construct confidence intervals for each, a certain percentage of those intervals would contain the true parameter.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It is calculated as half the width of the confidence interval, representing the maximum expected difference between the sample proportion and the true population proportion. In the given interval (0.512, 0.596), the margin of error can be found by subtracting the lower limit from the upper limit and dividing by two.

Sample Proportion

The sample proportion is the ratio of the number of successes in a sample to the total number of observations in that sample. It is denoted as 'p̂' and provides an estimate of the true population proportion. In the context of the confidence interval (0.512, 0.596), the sample proportion can be calculated as the midpoint of the interval, which gives a point estimate of the population proportion.