Decide whether applying the t -interval procedure to obtain...

Decide whether applying the $t$ -interval procedure to obtain a confidence interval for the population mean in question appears reasonable. Explain your answers. In the special report "Mousetrap: The Most-Visited Shoe and Apparel E-tailers" (Footwear News, Vol. $58,$ No. $3, p .18$ ), we found the following data on the average time, in minutes, spent per user per month from January to June of one year for a sample of 15 shoe and apparel retail Web sites. $$\begin{array}{|lcccc|} \hline 13.3 & 9.0 & 11.1 & 9.1 & 8.4 \\ 15.6 & 8.1 & 8.3 & 13.0 & 17.1 \\ 16.3 & 13.5 & 8.0 & 15.1 & 5.8 \\ \hline \end{array}$$

Decide whether applying the $t$ -interval procedure to obtain a confidence interval for the population mean in question appears reasonable. Explain your answers.
In the special report "Mousetrap: The Most-Visited Shoe and Apparel E-tailers"
(Footwear News, Vol. $58,$ No. $3, p .18$ ), we found the following data on the average time, in minutes, spent per user per month from January to June of one year for a sample of 15 shoe and apparel retail Web sites.
$$\begin{array}{|lcccc|}
\hline 13.3 & 9.0 & 11.1 & 9.1 & 8.4 \\
15.6 & 8.1 & 8.3 & 13.0 & 17.1 \\
16.3 & 13.5 & 8.0 & 15.1 & 5.8 \\
\hline
\end{array}$$

Key Concepts

Robustness of the t-Interval Method

The t-interval method is considered robust to some violations of the normality assumption when the sample size is moderate. However, if the underlying data are significantly skewed or contain outliers, the resulting confidence interval may not be reliable. Evaluating these aspects is critical when deciding whether the t-interval procedure is appropriate for the analysis.

t-Interval Procedure

The t-interval procedure is used to construct a confidence interval for the population mean when the population standard deviation is unknown. It relies on the t-distribution, which adjusts for the additional variability introduced by estimating the standard deviation from the sample. This method is particularly important in studies with small sample sizes where the central limit theorem may not assure normality of the sample mean.

Confidence Interval

A confidence interval is a range of values, derived from the sample statistics, that is likely to contain the true population parameter with a specified level of confidence. In this context, it gives an indication of where the true mean might lie, considering the sampling variability and uncertainty in the estimation process.

Assumptions of the t-Procedure

Applying the t-interval procedure requires certain assumptions such as the independence of observations and that the data are approximately normally distributed, especially when dealing with small sample sizes. These assumptions ensure that the t-distribution appropriately models the uncertainty in the sample mean.

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