Question

a national chain of department stores ranks its 1,100,000 salespeople by the monetary value of their sales. martinas sales are at the 45th percentile. pablos sales are at the 83rd percentile. (if necessary, consult a list of formulas.) (a) which of the following must be true about martinas sales? about 45% of the salespeople had sales lower in value than martina. the value of martinas sales were about 45% of the chains total. the value of martinas sales was in the top half of all of the salespeople. martina sold $4500 in merchandise. (b) which of the following must be true about martinas and pablos sales? both martina and pablo had sales higher in value than the median.

Explanation:

Step1: Recall percentile definition

The $k^{th}$ percentile of a data - set is a value such that $k\%$ of the data values are less than or equal to it.

Step2: Analyze Martina's percentile

Martina's sales are at the $45^{th}$ percentile. This means that about $45\%$ of the salespeople had sales lower in value than Martina.

Step3: Analyze option for Martina

• Option 1: Since Martina is at the $45^{th}$ percentile, about $45\%$ of the salespeople had sales lower in value than her. This is correct.
• Option 2: The percentile does not give the proportion of her sales to the chain's total. It gives the rank relative to other salespeople. So this is incorrect.
• Option 3: Since $45^{th}$ percentile is less than $50^{th}$ percentile (the median), her sales are not in the top - half. This is incorrect.
• Option 4: The percentile does not give the actual value of her sales. So this is incorrect.

Step4: Analyze Martina and Pablo's percentiles

The median is the $50^{th}$ percentile. Martina is at the $45^{th}$ percentile, so her sales are lower than the median. Pablo is at the $83^{rd}$ percentile, so his sales are higher than the median. So the statement "Both Martina and Pablo had sales higher in value than the median" is incorrect.

Answer:

(a) About 45% of the salespeople had sales lower in value than Martina.
(b) None of the given statements are true.