Formats of Confidence Intervals. In Exercises 9–12, express the confidence interval using the indicated format. (The confidence intervals are based on the proportions of red, orange, yellow, and blue M&Ms in Data Set 38 “Candies” in Appendix B.)


Green M&Ms Express 0.116 < p < 0.192 in the form of p +-E.

Verified step by step guidance

1. Identify the given confidence interval: 0.116 < p < 0.192. Here, p represents the population proportion, and the interval provides the range of plausible values for p.
2. To express the confidence interval in the form p ± E, calculate the midpoint of the interval. The midpoint is the point estimate (p̂) and can be found using the formula: $\mathrm{p̂}=\frac{\mathrm{Lower}+\mathrm{Upper}}{2}$, where 'Lower' is 0.116 and 'Upper' is 0.192.
3. Next, calculate the margin of error (E). The margin of error is the distance from the midpoint to either endpoint of the interval. Use the formula: $E=\mathrm{Upper}-\mathrm{p̂}$ or equivalently $E=\mathrm{p̂}-\mathrm{Lower}$.
4. Substitute the calculated midpoint (p̂) and margin of error (E) into the format p ± E. This will give the confidence interval in the desired form.
5. Verify your result by checking that the reconstructed interval (p̂ - E to p̂ + E) matches the original interval (0.116 to 0.192). This ensures the conversion was done correctly.

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 estimate, typically in the form of (lower limit, upper limit), and indicates the degree of uncertainty associated with a sample estimate. The width of the interval reflects the level of confidence, with wider intervals indicating more uncertainty.

Margin of Error (E)

The margin of error (E) quantifies the amount of random sampling error in a survey's results. It represents the maximum expected difference between the true population parameter and a sample estimate. In the context of confidence intervals, it is calculated as half the width of the interval, allowing for the expression of the interval in the form of p ± E, where p is the sample proportion.

Proportion

A proportion is a statistical measure that represents the fraction of a whole, often expressed as a decimal or percentage. In the context of the question, it refers to the proportion of a specific color of M&M's (e.g., green) in a sample. Understanding proportions is essential for calculating confidence intervals, as they serve as the basis for estimating population parameters.