STAT213 Midterm key

Question 1

a) P(less than 350K) = 57.84% from cumulative percent column

b) P(at least 200K) = 100% - 2.44% = 97.56%

c) P(at least 250K but less than 500K) = (208 + 160)/491 = 368/491 = 74.95%

d) Group mean = [187.5(12) + 225(64) + 300(208) + 425(160) + 550(47)]/491 = 352.1385. This puts the mean in the class 350K to under $500K. The median is in the class 250K to under 350K since the cumulative percent for this class is 57.84% and that of the previous class is 15.48%. Therefore, the mean is greater than the median and so the data would not be symmetric.

Question 2

a) P(at least 1) = 1 – P(0) = 1 – 0.96¹⁰ = 1 – 0.6648 = 0.3352 = 33.52%

b) P(0) = (15C0)(0.04⁰)(0.96¹⁵) = 0.5421

P(1) = (15C1)(0.04¹)(0.96¹⁴) = 0.3388

P(2) = (15C2)(0.04²)(0.96¹³) = 0.0988

P(no more than 2) = 0.9797 = 97.97%

c) µ = 100,000(0.04) = 4,000; s = sqrt(4000(0.96)) = 61.9677; µ - 3s = 4,000 – 3(61.9677) = 3,814 after rounding

Question 3

a) P(regular) = 0.71 P(sale | regular) = 0.332 P(regular and sale) = 0.23572

P(not regular) = 0.29 P(sale | not regular) = 0.824 P(not regular and sale) = 0.23896

P(sale) = 0.23572 + 0.23896 = 0.47468 = 47.47%

b) P(regular and sale) = 0.23572 P(>100 | regular and sale) = 0.564 P(regular and sale and >100) = 0.13294608

P(not regular and sale) = 0.23896 P(>100 | not regular and sale) = 0.252 P(not regular and sale and >100) = 0.06021792

P(sale and >100) = 0.13294608 + 0.06021792 = 0.193164

P(>100 | sale) = P(sale and >100)/P(sale) = 0.193164/0.47468 = 40.69%

c) We need P(sale | <100) and P(sale | >100). To find these, we build a crosstab:

	Sale	Not sale	Total
>100	0.193164	0.06028	0.253444
<100	0.281516	0.46504	0.746556
Total	0.47468	0.52532	1.0

Here is how the crosstab is built:

We have P(sale) = 0.47468 from part a.

We have P(sale and >100) = 0.193164 from part b.

We have P(<100) = 0.746556 from the initial information (second bullet point)

Everything else is derived using basic arithmetic.

P(sale | <100) = 0.281516/0.746556 = 37.71%

P(sale | >100) = 0.193164/0.253444 = 76.22%

Thus, the customer spending at least $100 is more likely to be visiting the store because of a sale.