1. In the year 2000, there were 125 major new automobile types available. If you look at the miles per gallon (MPG) for these vehicles, the distributions are roughly normal. The means and standard deviations are given by:
Mean Standard Deviation
City MPG 22.37 4.77
Highway MPG 29.09 5.46 a. The Lincoln Continental had a rating of 17 mpg city and 24 mpg highway.
Which rating, city or highway, is higher compared to the other cars?
Answer: Find and compare the Z-scores. The rating with the higher Z score is better.
City mpg z

17

22
4 .
77
.
37
Highway mpg z


24

29 .
09
5 .
46

1 .
13
 
.
93
The Lincoln has a higher rating for highway driving. b. The Saturn SL had a rating of 39 mpg city and 43 mpg highway. Which rating, city or highway, is higher compared to the other cars?
Answer: Find and compare the Z-scores. The rating with the higher Z score is better.
39

City mpg z

22
4 .
77
.
37

3 .
49
Highway mpg z

43

29 .
09
5 .
46

2 .
55
The Saturn has a higher rating for city driving.
1

c. The Mazda 626 had a city rating of 26 mpg. What highway rating would it have so that the city and highway ratings were the same compared to other cars?
Answer: You need to think about this one. First find a z score for 26 mpg.
City mpg z

26

22
4 .
77
.
37

.
76
Next, set the z score for the highway mpg equal to .76 and solve for x
Highway mpg z
 x

29 .
09

.
76 x

33 .
24
5 .
46
For the following problems, draw and shade the region you are looking for or are given.
2. On a typical Saturday, Genuardis Markets reports that the mean amount of money spent by customers is $27.21 and a standard deviation of $7.93. The distribution is roughly normal. a. The middle 95% of customers spent between what two amounts?
Find the z values associated with the middle 95%

1 .
96
 z

1 .
96
Find the dollar amounts that correspond to these values.

1 .
96
 x

7 .
27
93
.
21
 x

1 .
96
 x

27
7 .
93
.
21
$ 11 .
67
 x

$ 42 .
75
2

b. If a person spends $10, in what percentile does that place her?
Find the probability of $10 and convert to a percent z

10

27
7 .
93
.
21 z
 
2 .
17 1.5 percentile c. If a person spends $40, she is in the top ______ percent of spending?
Find the probability of $40 and convert to a percent z

40

27 .
21
7 .
93 z

1 .
61 5.37% d. What range of amounts do the middle 60% represent?
Find the values associated with the middle 60%

.
84
 z

.
84
Find the dollar amounts that correspond to these values.

.
84
 x

7 .
27
93
.
21
 x

.
84
 x

27
7 .
93
.
21
$ 20 .
55
 x

$ 33 .
87
3

e. If a customer spends in the top 2%, how much does he spend?
Find the z score corresponding to the top 2% then solve for x
2 .
05
 x

27 .
21
7 .
93 x

$ 43 .
47 f. If a customer spends in the 99.5
th percentile, how much does he spend?
Find the z score corresponding to 99.5 percentile then solve for x
2 .
58
 x

27 .
21
7 .
93 x

$ 47 .
67 g. If a customer spends in the 99.99
th percentile, how much does he spend?
Find the z score corresponding to 99.99 percentile then solve for x
3 .
72
 x

27
7 .
93
.
21 x

$ 56 .
71
4

Over a ten year period, American League baseball players and National League players had the following averages and standard deviations. Averages over that long a period of time would be essentially normal.
Mean Standard Deviation
American League
National League
.263
.254
.202
.214 a. Jim Thome played in both leagues and had a batting average of .277 in the
American League and .273 in the National League. Show work to determine what league his batting average was higher compared to the rest of the league.
Answer: Find and compare the Z-scores. The batting average with the higher Z score is better.
.
277

.
263
American League z
 
.
07
.
202
National League z

.
273

.
254
.
214

.
09
His batting average was better in the National League. b. Aaron Rowand played in both leagues and had a batting average of .250 in the
American League and .241 in the National League. Show work to determine what league his batting average was higher compared to the rest of the league.
American League z

.
250

.
263
.
202
 
.
0644
National League z

.
241

.
254
.
214
 
.
0607
His average was slightly better in the National League.
5

c. If a player plays in both leagues and bats .300 in the National League, show appropriate work to determine what average he would have to have in the
American League so that his average would be the same compared to the rest of the league.
Answer: You need to think about this one. First find a z score for .300. z

.
300

.
254
.
214

.
21
Next set the z score for the American League equal to .21 and solve for x.
.
21
 x

.
254 x
.
214

.
299
For the following problems, draw and shade the region you are looking for or are given. At a busy traffic intersection, the mean waiting time to go through the intersection is 3 minutes, 20 seconds with a standard deviation of 52 seconds.
(Convert everything to seconds) a. The middle 95% of drivers spent between what two amounts of waiting time (1 decimal place)?
Find the z values associated with the middle 95%

1 .
96
 z

1 .
96
Find the times that correspond to these values.

1 .
96
 x

200
52
 x

1 .
96
 x

200
52
98 .
1
 x

301 .
9
6

b. If a driver waits exactly two minutes, what percentile does that represent?
(Assume the longer the wait, the higher the percentile)
Find a z score and the corresponding probability for 2 minutes. z

120

52
200
 
1 .
54 6.18 percentile c. If a person waits 6 minutes he is in the top ______ percent of waiting?
Find a z score and the corresponding probability for 6 minutes. z

360

52
200

3 .
08 .1% d. What percentage of drivers waits between 2 and 4 minutes?
Find the z scores and probabilities associated with these times. z

120

200
52
 z
 z

240

200
52

1 .
54
 z

.
0618
 p ( x )
.
77

.
7794 .
7794

.
0618

.
7176
71.76%
7

e. If a driver waits in the top 2%, how much time does he wait?
Find a z score associated with the top 2% and solve for the time.
2 .
05
 x

200
52 x

306 .
6 seconds or more f. If a customer waits in the 0.5 percentile of waiting times, how much times does he spend waiting?
Find a z score associated with .5% and solve for the time.

2 .
58
 x

200
52 x

65 .
8 seconds or more
8