Constructing Confidence Intervals Use the data set to (a)...

Constructing Confidence Intervals Use the data set to (a) find the sample mean, (b) find the sample standard deviation, and (c) construct a $99 \%$ confidence interval for the population mean. Assume the population is normally distributed. If convenient, use technology. The weekly time (in hours) spent weight lifting for 16 randomly selected college football players $$\begin{array}{llllllll} 7.4 & 5.8 & 7.3 & 7.0 & 8.9 & 9.4 & 8.3 & 9.3 \\ 6.9 & 7.5 & 9.0 & 5.8 & 5.5 & 8.6 & 9.3 & 3.8 \end{array}$$

Constructing Confidence Intervals Use the data set to (a) find the sample mean, (b) find the sample standard deviation, and (c) construct a $99 \%$ confidence interval for the population mean. Assume the population is normally distributed. If convenient, use technology.
The weekly time (in hours) spent weight lifting for 16 randomly selected college football players
$$\begin{array}{llllllll}
7.4 & 5.8 & 7.3 & 7.0 & 8.9 & 9.4 & 8.3 & 9.3 \\
6.9 & 7.5 & 9.0 & 5.8 & 5.5 & 8.6 & 9.3 & 3.8
\end{array}$$

Key Concepts

Sample Mean

The sample mean is the arithmetic average of the data and serves as an estimate of the population mean. It is calculated by summing all observations and dividing by the number of observations, thus providing a measure of central tendency for the data set.

Sample Standard Deviation

The sample standard deviation quantifies the dispersion or spread of data points around the sample mean. It is computed by first determining the squared deviations from the mean, averaging these for a sample variance, and then taking the square root, offering insight into the variability within the sample.

Confidence Interval

A confidence interval is a range derived from sample data which is used to estimate a population parameter, such as the mean, with a certain level of confidence (e.g., 99%). It reflects the uncertainty inherent in sampling and provides a probabilistic measure of where the true population parameter is likely to be located.

t-Distribution

The t-distribution is utilized when the population standard deviation is unknown and the sample size is small. It adjusts for additional uncertainty by having heavier tails than the normal distribution, and is essential in constructing confidence intervals for the population mean under the assumption of an underlying normal distribution.

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