Question

Small Sample Size and Outliers As we have seen, bootstrap distributions are generally symmetric and bell-shaped and centered at the value of the original sample statistic. However, strange things can happen when the sample size is small and there is an outlier present. Use StatKey or other technology to create a bootstrap distribution for the standard deviation based on the following data: \(8 \quad 10\) 72 \(13 \quad 8\) \(\begin{array}{ll}10 & 50\end{array}\) Describe the shape of the distribution. Is it appropriate to construct a confidence interval from this distribution? Explain why the distribution might have the shape it does.

Short answer

The shape of the bootstrap distribution might be skewed or irregular due to the outlier in the data set. Depending on the observed skewness or irregularity, it might not be appropriate to construct a confidence interval from this distribution. The presence of an outlier (72) in the small data set is what causes the particular shape of the distribution - being substantially distant from the other data points, it significantly influences the standard deviation whenever it is included in a sample.

Step-by-step solution

Generating the Bootstrap Distribution

Firstly, with technology tools like StatKey or others reflecting statistical modelling, generate a bootstrap distribution for the standard deviation of the given data: \(8, 10, 72, 13, 8, 10, 50\). The bootstrap distribution is created by taking numerous samples from the data, with replacement, and calculating the statistic (in this case, standard deviation) from each sample.

Observing the Shape of the Distribution

Next is to observe the shape or pattern of the bootstrap distribution. It might not follow a symmetric, bell-shaped curve, mainly due to the presence of an outlier in the small data set.

Discussing the Appropriateness of a Confidence Interval

Analyze whether it is appropriate to construct a confidence interval from this distribution. If the bootstrap distribution is not symmetric, or if it is substantially skewed, it may not be suitable to form a confidence interval from it since standard methods often require the sampling distribution to be approximately symmetric.

Understanding the Shape of Distribution

Explain why the distribution might have the shape it does. This can largely be due to the presence of an outlier (72) in the data set. Being a single data point that lies far from the other values, the outlier would have a significant impact on the standard deviation whenever it is included in a sample, thereby influencing the shape of the bootstrap distribution.