Question

a national chain of department stores ranks its 1,000,000 salespeople by the monetary value of their sales. lucys sales are at the 56th percentile. (a) which of the following must be true about lucys sales? o about 28% of the salespeople had sales lower in value than lucy. o the value of lucys sales were about 28% of the chains total. o the value of lucys sales was in the top half of all of the salespeople. o lucy sold $2800 in merchandise. (b) which of the following must be true about lucys and austins sales? o both lucy and austin had sales higher in value than the median. o both lucy and austin had sales lower in value than the median. o austins sales were higher in value than lucys sales. o lucys sales were higher in value than austins sales.

Explanation:

Step1: Recall percentile definition

The $k^{th}$ percentile of a data - set is a value such that $k$ percent of the data values are less than or equal to it. Lucy is at the 56th percentile, which means 56% of the salespeople have sales less than or equal to Lucy's sales, and 44% of the salespeople have sales greater than Lucy's sales.

Step2: Analyze option for part (a)

• Option 1: Since Lucy is at the 56th percentile, about 56% of the salespeople had sales lower in value than Lucy, not 28%, so this is false.
• Option 2: There is no information to suggest that Lucy's sales are 28% of the chain's total. Percentile is about rank, not proportion of total sales, so this is false.
• Option 3: Since 56% > 50%, Lucy's sales are in the top - half of all salespeople. This is true.
• Option 4: There is no information given about the actual dollar amount of Lucy's sales, so this is false.

Step3: Analyze part (b) (assuming Austin is at a lower percentile, say 40th percentile for the sake of analysis, since no info about Austin is given in the problem statement but we can still analyze based on percentile concepts)

The median is the 50th percentile. Lucy is at the 56th percentile (above the median) and if Austin is at a lower percentile (say 40th percentile), then Lucy's sales were higher in value than Austin's sales.

• Option 1: We don't know Austin's percentile relative to 50% well - enough to say they both have sales higher than the median.
• Option 2: We know Lucy is above the median, so this is false.
• Option 3: Since Lucy is at 56th percentile and assuming Austin is at a lower percentile, this is false.
• Option 4: Since Lucy is at a higher percentile than Austin (assuming a lower percentile for Austin), Lucy's sales were higher in value than Austin's sales. This is true.

Answer:

(a) The value of Lucy's sales was in the top half of all of the salespeople.
(b) Lucy's sales were higher in value than Austin's sales.