A clinic offers a weight-loss program. A review of its...

A clinic offers a weight-loss program. A review of its records found the following amounts of weight loss, in pounds, for a random sample of 24 of its clients at the conclusion of a 4 -month program: $$ \begin{array}{llllllll} 18 & 25 & 16 & 11 & 15 & 20 & 16 & 19 \\ 28 & 25 & 26 & 31 & 45 & 40 & 36 & 19 \\ 28 & 25 & 36 & 16 & 35 & 20 & 16 & 19 \end{array} $$ a. Find a $99 \%$ confidence interval for the population mean. b. Without doing the calculations, explain whether a $90 \%$ confidence interval for the population mean would be wider than, narrower than, or the same as that found in part a. would be wider than, narrower than, or the same as that found in part a.

A clinic offers a weight-loss program. A review of its records found the following amounts of weight loss, in pounds, for a random sample of 24 of its clients at the conclusion of a 4 -month program:
$$
\begin{array}{llllllll}
18 & 25 & 16 & 11 & 15 & 20 & 16 & 19 \\
28 & 25 & 26 & 31 & 45 & 40 & 36 & 19 \\
28 & 25 & 36 & 16 & 35 & 20 & 16 & 19
\end{array}
$$
a. Find a $99 \%$ confidence interval for the population mean.
b. Without doing the calculations, explain whether a $90 \%$ confidence interval for the population mean would be wider than, narrower than, or the same as that found in part a. would be wider than, narrower than, or the same as that found in part a.

Key Concepts

Confidence Level and Its Effect on Interval Width

The confidence level represents the probability that the confidence interval will contain the true population parameter. A higher confidence level requires a larger critical value, which in turn results in a wider confidence interval, reflecting greater uncertainty in the point estimate.

Critical Value

The critical value is the threshold value from a probability distribution (such as the t-distribution) that corresponds to a specified level of confidence. It is used to scale the standard error in order to create the margin of error for the confidence interval.

Sample Mean and Sample Standard Deviation

The sample mean provides an unbiased estimate of the population mean, while the sample standard deviation measures the dispersion of the data. These statistics are fundamental in constructing a confidence interval for the population mean, especially when using the t-distribution.

t-Distribution

The t-distribution is used when estimating population parameters from a small sample size or when the population standard deviation is unknown. It differs from the normal distribution by having heavier tails, which appropriately adjusts for the additional uncertainty in the estimate.

Confidence Interval

A confidence interval is a range of values, derived from the sample statistics, that is likely to contain the true population parameter. It accounts for sample variability and uncertainty, and is typically constructed using the sample mean, sample standard deviation, and a critical value from an appropriate probability distribution.

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