(a)(1) Choose, revise, and finally 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) State what your data might represent with an applicable unit; state the 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 selected for your data. The unit must accompany the value when reported.

(2) What would be an estimate for the variance based on your estimate for the standard deviation? Do not forget its unit; it is not the same unit selected.

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

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

(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, would the variance and standard deviation increase, decrease, or remain unchanged? Why? Briefly explain your answer.

(e)Provide the five-number summary.

(f)Determine the Interquartile Range (IQR).

(g)Using your "actual value" for the standard deviation (SD) from Part (d)(1) above, calculate actually what percentage of your 12 data points would fall within one SD of the mean, i.e., within the interval (mean - SD, mean + SD).Compare your calculated answer with the Empirical Rule percentage.Briefly explain the consistency or inconsistency observed and the reason for it.