STAT213 Worksheet #2 Solutions

 

First, we’ll make up a crosstab:

Bay

Not Bay

Total

Walmart

7

33

40

Not Walmart

8

52

60

Total

15

85

100

1)      P(Bay or Walmart) = P(Bay) + P(Walmart) - P(Bay and Walmart) = 15% + 40% - 7% = 48%

2)      P(Walmart | Bay) = P(Bay and Walmart)/P(Bay) = 7/15 = 46.67%

3)      P(Walmart and not Bay) = 33% from the crosstab

4)      P(not Bay | Walmart) = P(Walmart and not Bay)/P(Walmart) = 33/40 = 82.5%

5)      P(not Walmart | not Bay) = P(not Bay and not Walmart)/P(not Bay) = 52/85 = 61.18%

6)      P(neither Bay nor Walmart) = 52% from crosstab. The other solution is to use DeMorgan’s Law: P(neither Bay nor Walmart) = 100% - P(Bay or Walmart) = 100% - 48% = 52%.

7)      P(Bay | Walmart) = 7/40 = 17.5%; P(Bay | not Walmart) = 8/60 = 13.33%. So, someone who shops at Walmart is more likely to shop at The Bay than someone who doesn’t shop at Walmart.

8)      We have P(Bay and Walmart) = 7%. P(Bay)*P(Walmart) = (0.15)(0.4) = 0.06 = 6%. Since these are not equal, the events are not independent and so shopping at one store depends on shopping at the other. One other solution is to note that P(Bay | Walmart) = 17.5% from question 7. However, P(Bay) = 15%. Since P(Bay) doesn’t equal P(Bay | Walmart), we reach the same conclusion.

9)      Since P(Bay and Walmart) = 7% is not zero, the events of shopping at the 2 stores are not mutually exclusive.

10)  P(revenue ³ 500K) = (438 + 365)/1000 = 803/1000 = 80.3%

11)  P(profit ³ 100K | revenue ³ 500K) = (81 + 98 + 11 + 67)/(438 + 365) = 257/803 = 32%

12)  P(profit < 100K) = (268 + 473)/1000 = 741/1000 = 74.1%

13)  P(revenue < 500K  | profit < 100K) = (141 + 54)/(268 + 473) = 195/741 = 26.32%

14)  Revenue under $500,000 and net profit of $500,000 or more are mutually exclusive since the crosstab value is zero. These categories must be mutually exclusive since you can’t have a net profit of $500,000 or more if your revenue is less than that.

15)  P(revenue ³ 500K or profit ³ 100K) = P(revenue ³ 500K) + P(profit ³ 100K) – P(revenue ³ 500K and profit ³ 100K) = (438 + 365)/1000 + (181 + 78)/1000 – (81 + 98 + 11 + 67)/1000 = 805/1000 = 80.5%. The alternate solution is P(revenue ³ 500K or profit ³ 100K) = 100% - P(revenue < 500K and profit < 100K) = 100% - (141 + 54)/1000 = 100% - 19.5% = 80.5%.

 

16)  This table summarizes the results:

Profit|Revenue

< $500K

$500K - $1000K

$1000K +

$100K - $500K

2

81

98

$500K +

0

11

67

Total

197

438

365

 

1.02%

21.00%

45.21%

For example, for revenue under $500,000, the percentage of these companies with net profit of $100,000 or more is (2 + 0)/197 = 1.02%. As revenues increase, the percentage of companies with net profit of $100,000 or more increases.

17)  P(MC) = 0.352 P(stay | MC) = 0.642 P(MC and stay) = 0.225984

P(no MC) =0.648 P(stay | no MC) = 0.124 P(no MC and stay) = 0.080352

P(stay) = 0.306336 which converts to 30.63%

18)  P(MC and stay) = 0.225984 P(surgery | MC and stay) = 0.042

P(MC and stay and surgery) = 0.009491328

P(no MC and stay) = 0.080352 P(surgery | no MC and stay) = 0.058

P(no MC and stay and surgery) = 0.004660416

P(stay and surgery) = 0.014151744

We need P(no surgery | stay).  There are two ways to solve this.

Method 1: P(surgery | stay) = 0.014151744/0.306336 = 0.0462 = 4.62% Then P(no surgery | stay) = 100% = 4.62% = 95.38%.

Method 2: P(stay and no surgery) = P(stay) – P(stay and surgery) = 0.306336 - 0.014151744 = 0.292184256. Then P(no surgery | stay) = 0.292184256/0.306336 = 0.9538 = 95.38%.

 

 

From the textbook

 

Section 3-3

Question 9

This is a situation in which it is easier to construct a crosstab with raw frequencies.

 

18-21

22-29

Total

Respond

73

255

328

Not respond

11

20

31

Total

84

275

359

P(18-21 or not respond) = 84/359 + 31/359 – 11/359 = 104/359 = 28.97%

 

Question 10

P(18-21 or respond) = 84/359 + 328/359 – 73/359 = 339/359 = 94.43%

 

Question 13

P(copier or printer) = 25% + 50% - 13% = 62%

 

Question 14

P(neither copier nor printer) = 100% - 62% = 38%

 

Question 29

a.

 

Inner city

Suburb

Outside

Total

Yes

8.5

35.8

0.9

45.2

No

1.8

27.6

2.9

32.3

Undecided

2.3

15.9

4.3

22.5

Total

12.6

79.3

8.1

100

b. P(city or not yes) = P(city) + P(not yes) – P(city and not yes) = (12.6% + 79.3%) + (32.3% + 22.5%) – (1.8% + 27.6% + 2.3% + 15.9%) = 99.1%

c. P(city or not yes) = 100% - P(outside and yes) = 100% - 0.9% = 99.1%

 

Section 3-4

Question 3

a.

 

Import

Not import

Total

Over 5

9

19

28

Not over 5

34

38

72

Total

43

57

100

   b. P(over 5 | import) = 9/43 = 0.2093 = 20.93%

   c. P(import | not over 5) = 34/72 = 0.4722 = 47.22%

   d. P(not over 5 | not import) = 38/57 = 0.6667 = 66.67%

   e. Left side = P(import and over 5) = 0.09

   Right side = P(import)P(over 5) = (0.43)(0.28) = 0.1204

   Left side does not equal right side.

   Therefore, owning a vehicle over 5 years old depends on it being an import.

 

Question 6

a.

 

Own return

Not own return

Total

Refund

68

8

76

No refund

17

7

24

Total

85

15

100

b. P(refund | own return) = 68/85 = 0.8 = 80%

P(refund | not own return) = 8/15 = 0.5333 = 53.33%

Those who do their own return are more likely to receive a refund.

 

Section 3-5

Question 17

a. P(income ³ 60K) = (0.246)(0.702) + (0.754)(0.354)  = 0.172692 + 0.266916 = 0.439608 = 43.96%

b. P(income < 60K) = 100% - 43.9608% = 56.0392%.

 

University

No university

Total

60K+

17.2692

26.6916

43.9608

<60K

7.3308

48.7084

56.0392

Total

24.6

75.4

100

P(no university | <60K) = 48.7084/56.0392 = 0.8692 = 86.92%

Question 20

a. P(recycle) = (0.45)(0.12) + (0.3)(0.18) + (0.25)(0.3) = 0.183 = 18.3%.

b. P(recycle milk jugs) = (0.45)(0.12)(0.04) + (0.3)(0.18)(0.08) + (0.25)(0.3)(0.4) = 0.03648 = 3.65%.

c. P(recycle milk jugs | recycle) = 3.648/18.3 = 19.93%