Problem II - [18 points]

(a) (1) Choose and write down a sample of 12 distinct (different) positive integers (no modes), each less than 100, in a way that your data set would have a range of 90, a mean of 59, and a median of 55

(2) In order not to treat the data as an abstract set, state what your data might represent with an applicable unit.

(b) Show your data set and your work to demonstrate that your data set does have the statistical characteristics mentioned. Consider your data set as a sample.

(c) (1) Estimate the standard deviation using the "range rule of thumb": four standard deviations practically cover the span of the data (about 95% for normal distributions), when rank ordered. Do not calculate the actual standard deviation value using its formula or technology for this part. The standard deviation found would have the same unit as in Part (a) above; the unit should accompany the value when reported.

(2) What would be an estimate for the variance? Do not forget its unit.

(d) Use technology (for example, the "Variance and Standard Deviation Applet," provided under

Content > Course Resources > Free Web Applets in LEO, Excel, TI calculator) to:

(1) find the actual values (in two decimal places) for the variance (with its unit) and the standard deviation (with its unit) of your data set, considered as a sample.

(2) If you mistakenly treated your sample as a population, what would be the impact of your mistake on calculated values for the variance and the standard deviation? Why? Briefly explain.

(e) Provide the five-number summary.

(f) Determine the Interquartile Range (IQR).

(g) Using your "actual value" for the standard deviation (STD) from Part (d)(1) above, determine what percentage of your data points would fall within one STD of the mean, i.e., within the interval (mean - STD, mean + STD). Compare your answer with the Empirical Rule percentage. Briefly explain the inconsistency, if any.