In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.
c = 0.90, s = 25.6, n = 16, xbar = 72.1

검증된 단계별 안내

Step 1: Identify the given values and understand the problem. Here, c = 0.90 (confidence level), s = 25.6 (sample standard deviation), n = 16 (sample size), and x̄ = 72.1 (sample mean). The goal is to calculate the margin of error and construct the confidence interval using the t-distribution.

Step 2: Calculate the degrees of freedom (df) for the t-distribution. The formula is df = n - 1. Substitute n = 16 into the formula to find df.

Step 3: Determine the critical t-value (t*) for the given confidence level (c = 0.90) and degrees of freedom (df). Use a t-distribution table or statistical software to find t* corresponding to a two-tailed test with a 90% confidence level.

Step 4: Compute the margin of error (E) using the formula E = t* × (s / √n). Substitute the values of t*, s = 25.6, and n = 16 into the formula. Simplify the expression to find E.

Step 5: Construct the confidence interval for the population mean (μ) using the formula: Confidence Interval = x̄ ± E. Substitute x̄ = 72.1 and the calculated margin of error (E) into the formula to express the confidence interval.

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주요 개념

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Margin of Error

The margin of error quantifies the uncertainty in a sample estimate. It is calculated using the formula: Margin of Error = t * (s / √n), where t is the t-score corresponding to the desired confidence level, s is the sample standard deviation, and n is the sample size. A smaller margin of error indicates a more precise estimate of the population parameter.

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the population parameter with a specified level of confidence. It is constructed using the formula: Confidence Interval = x̄ ± Margin of Error, where x̄ is the sample mean. For a 90% confidence level, the interval provides a range where we expect the true population mean to fall.

t-Distribution

The t-distribution is a probability distribution used when estimating population parameters when the sample size is small (typically n < 30) and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the increased variability in smaller samples. The t-score is used in calculating the margin of error and confidence intervals.

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.c = 0.90, s = 25.6, n = 16, xbar = 72.1