In Exercises 5–8, (a) identify the critical value ta/2 used for finding the margin of error, (b) find the margin of error, (c) find the confidence interval estimate of u, and (d) write a brief statement that interprets the confidence interval.


Pepsi Weights Here are summary statistics for the weights of Pepsi in randomly selected cans: n=36, x=0.82410 lb, s=0.00570 lb (based on Data Set 37 “Cola Weights and Volumes” in Appendix B). Use a confidence level of 99%.

Verified step by step guidance

1. Step 1: Identify the critical value (tα/2). To find the critical value for a 99% confidence level, use the t-distribution table. The degrees of freedom (df) are calculated as n - 1, where n is the sample size. Here, df = 36 - 1 = 35. Look up the t-value corresponding to a 99% confidence level (α = 0.01, so α/2 = 0.005) and df = 35.
2. Step 2: Calculate the margin of error (E). The formula for the margin of error is E = tα/2 * (s / √n), where tα/2 is the critical value, s is the sample standard deviation, and n is the sample size. Substitute the values: s = 0.00570, n = 36, and the tα/2 value found in Step 1.
3. Step 3: Find the confidence interval estimate of μ. The confidence interval is given by the formula: CI = x̄ ± E, where x̄ is the sample mean and E is the margin of error. Substitute x̄ = 0.82410 and the margin of error calculated in Step 2.
4. Step 4: Write the confidence interval in interval notation. Express the confidence interval as (lower bound, upper bound), where the lower bound is x̄ - E and the upper bound is x̄ + E.
5. Step 5: Interpret the confidence interval. Write a brief statement explaining that we are 99% confident that the true population mean weight of Pepsi cans lies within the calculated confidence interval.

Key Concepts

Critical Value (t-distribution)

The critical value, denoted as tα/2, is a point on the t-distribution that corresponds to a specified confidence level. It is used to determine the margin of error in confidence intervals, especially when the sample size is small or the population standard deviation is unknown. For a 99% confidence level, this value is found using statistical tables or software, reflecting the area in the tails of the distribution.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It is calculated by multiplying the critical value by the standard error of the sample mean. This value indicates the range within which the true population parameter is expected to lie, providing a buffer around the sample mean to account for sampling variability.

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 constructed using the sample mean and the margin of error, providing an interval estimate rather than a point estimate. The width of the interval reflects the level of confidence; a higher confidence level results in a wider interval, indicating greater uncertainty about the exact value of the parameter.