Download Chapter 6: The Standard Deviation as a Ruler and the Normal Model

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AP Statistics: Chapter 6 - Normal Percentiles
1.
Suppose the class took a 40-point quiz. Results show that a mean score of 30, median 32, IQR
8, standard deviation 6, min 12 and lower quartile 27. You got a 35. What happens to each of
the statistics if…
a. I decide to weight the quiz as 50 points by adding 10 points to each score. Your
score is now 45.
x=
M=
IQR =
s=
min =
Q1=
b. I decide to weight the quiz as 80 points by doubling each score. Your score is now
70.
x=
M=
IQR =
s=
min =
Q1=
c.
I decide to count the quiz as 100 points by doubling each score, then adding 20
points. Your score is now 90.
x=
M=
IQR =
s=
min =
Q1=
2. Consider the three athletes’ performances shown below in a 3-event competition. Note that
each person finishes first, second, and third one time. Who deserves the gold medal? And who
gave the most remarkable performance of the competition?
Competitor
A
B
C
Mean
St Dev
100 m Dash
10.1 sec
9.9 sec
10.3 sec
10 sec
0.2 sec
Event
Shot Put
66’
60’
63’
60’
3’
Long Jump
26’
27’
27’ 3”
26’
6”
More Percentile Practice:
3. Consider the Angus weights model N(1152, 84).
a) What percent of steers weigh over 1250lbs?
b) What percent of steers weigh under 1200 lbs?
c) What percent of steers weigh between 1000 and 1100 lbs?
d) What weight represents the 40th percentile?
e) What weight represents the 99th percentile?
f) What is the IQR of the weights of these Angus steers?
4. Companies that design furniture for elementary school classrooms produce a variety of
sizes for kids of different ages. Suppose the heights of kindergarten children can be
modeled using N(38.2, 1.8).
a) What fraction of kindergarten kids should the company expect to be less than 36 in. (3 ft)
tall?
b) In what height interval should the company expect to find the middle 80% of
kindergarteners?
c) At least how tall are the biggest 10% of kindergarteners?
AP Statistics: Chapter 6 - Normal Percentiles (Key)
1.
Suppose the class took a 40-point quiz. Results show that a mean score of 30, median 32,
IQR 8, standard deviation 6, min 12 and lower quartile 27. You got a 35. What happens to
each of the statistics if…
a. I decide to weight the quiz as 50 points by adding 10 points to each score. Your
score is now 45.
x = 40
M = 42
IQR = 8
s= 6
min = 12
Q1= 37
b. I decide to weight the quiz as 80 points by doubling each score. Your score is now
70.
x = 60
M = 64
IQR = 16
s = 12
min = 24
Q1= 54
c.
I decide to count the quiz as 100 points by doubling each score, then adding 20
points. Your score is now 90.
x = 80
M = 84
IQR = 16
s = 12
min = 44
Q1= 74
2. Consider the three athletes’ performances shown below in a 3-event competition. Note that
each person finishes first, second, and third one time. Who deserves the gold medal? And
who gave the most remarkable performance of the competition?
Competitor
A
B
C
Mean
St Dev
100 m Dash
10.1 sec
9.9 sec
10.3 sec
10 sec
0.2 sec
Event
Shot Put
66’
60’
63’
60’
3’
Long Jump
26’
27’
27’ 3”
26’
6”
We want to compare athletic performance across several different events, so we need
some way to compare without units. We will compare z-scores from each event.
Event z-scores
Total zscores
Competitor
100 m Dash
Shot Put
Long Jump
A
0.5
2
0
1.5
B
-0.5
0
2
2.5
C
1.5
1
2.5
2
Note that the 100-m Dash should be reversed: the lower the time, the better, so a
negative z-score is better. For computational purposes, the signs of the z-scores should
be reversed before adding to the other 2 z-scores. Therefore, overall, competitor B
deserves the gold medal. Competitor C gave the most remarkable performance in the Long
Jump (has the highest z-score).
More Percentile Practice:
3. Consider the Angus weights model N (1152, 84).
a) What percent of steers weigh over 1250lbs?
12.17%
b) What percent of steers weigh under 1200 lbs?
71.61%
c) What percent of steers weigh between 1000 and 1100 lbs?
23.28%
d) What weight represents the 40th percentile?
x = 1130.7188 lbs
e) What weight represents the 99th percentile?
x = 1347.4132 lbs
f) What is the IQR of the weights of these Angus steers?
IQR = 113.3142 lbs
4. Companies that design furniture for elementary school classrooms produce a variety of
sizes for kids of different ages. Suppose the heights of kindergarten children can be
modeled using N (38.2, 1.8).
a) What fraction of kindergarten kids should the company expect to be less than 36 in. (3 ft)
tall?
11.08%
b) In what height interval should the company expect to find the middle 80% of
kindergarteners?
Between 35.89 in. to 40.51 in.
c) At least how tall are the biggest 10% of kindergarteners?
At least 40.5068 in.