Download Use the Normal Probability Distribution to make decisions about a

NORMAL PROBABILITY DISTRIBUTION
Objective: Use the Normal Probability Distribution to make decisions about a
population.
Scenario: An apparel company makes blue jeans and leather pants.
I. The female division.
Female Data in inches
66.4
66.1
66.8
64.7
68.1
69.2
63.6
66.3
66.7
64.9
67.5
64.2
67.9
67.6
60.2
62.2
63.1
57.6
69.4
64.3
67.8
65.1
68.4
67.2
66.1
66.7
62.2
63.2
68.9
67.8
67.2
58.1
A. Use the "Statistics" function of your calculator to find the sample mean and
sample standard deviation for the data. (Round to tenths.)
1. Mean
2. Standard deviation
Use these statistics (sample mean and sample standard deviation) as point
estimates for the population parameters (population mean and population standard
deviation) when calculating the standard (z) scores.
B. Use the Normal Probability Distribution table or the built-in functions of
your calculator to find:
1. What percent of female adults are taller than 6 feet (72 inches)?
2. What percent of female adults are taller than 5 feet (60 inches)?
3. What percent of female adult heights are between 60 inches and 72
inches?
C. Because of the high cost of leather, the company has decided they cannot
profitably make leather pants in all sizes.
Use the Normal Probability Distribution table or the built-in functions of
your calculator to find the height corresponding to the following percentages.
These are the heights of the shortest and tallest females who can purchase leather
pants from this company.
1. The bottom 8%
2. The upper 6%
II. The male division
Male Data in inches
68
67.7
70.5
66.3
65.5
65.1
66.7
69.3
68.1 72.5 65.4 71.2
65.3 65.5 72 73.2
67.5 70.2 67.4 71.8
67.7 67 73.8 66.5
67.7
62.5
65.1
66.1
73.5
77.2
67.2
68.6
A. Use the "Statistics" function of your calculator to find the sample mean and
sample standard deviation
for the data. (Round to hundredths.)
1. Mean
2. Standard deviation
Use these statistics (sample mean and sample standard deviation) as point
estimates for the population
parameters (population mean and population standard deviation) when
calculating the standard (z) scores.
B. Use the Normal Probability Distribution table or the built-in functions of
your calculator to find:
1. What percent of male adults are shorter than 6 feet (72 inches)?
2. What percent of male adults are shorter than 5 feet (60 inches)?
3. What percent of male adult heights are between 60 inches and 72 inches?
C. Because of the high cost of leather, the company has decided they cannot
profitably make leather pants in all sizes.
Use the Normal Probability Distribution table or the built-in functions of
your calculator to find the heights corresponding to the following percentages.
These are the heights of the shortest and tallest males who can purchase leather
pants from this company.
1. The bottom 9%
2. The upper 7%
NORMAL PROBABILITY DISTRIBUTION
Objective: Use the Normal Probability Distribution to make decisions about
a population.
I. The female division
Female Data in inches
66.4
66.1
66.8
64.7
68.1
69.2
63.6
66.3
66.7
64.9
67.5
64.2
67.9
67.6
60.2
62.2
63.1
57.6
69.4
64.3
67.8
65.1
68.4
67.2
66.1
66.7
62.2
63.2
68.9
67.8
67.2
58.1
A. Using the calculator to find the sample mean and standard deviation (rounded
to tenths) :
1. Mean height of an adult female is 65.5 inches.
2. Standard deviation of an adult female is 3 inches.
B1. What percent of female adults are taller than 6 feet?
Using a single page Standard Normal Probability Distribution Table starting
at the mean and extending to z.
z = 2.17 has a probability of 0.4850
0.5000 – 0.4850 = 0.0150 = 1.5%
Using a two page Standard Normal Probability Distribution Table starting at z
and extending to the left.
z = 2.17 has a probability of 0.9850
1.0000 – 0.9850 = 0.0150 = 1.5%
About 1.5% of females are taller than 6 feet.
B2. What percent of female adults are taller than 5 feet?
Using a single page Standard Normal Probability Distribution Table starting
at the mean and extending to z.
z = – 1.83 has a probability of 0.4664
0.5000 + 0.4664 = 0.9664 ≈ 96.6%
Using a two page Standard Normal Probability Distribution Table starting at
z and extending to the left.
z = – 1.83 has a probability of 0.0336
1.0000 – 0.0336 = 0.9664 ≈ 96.6%
About 96.6% of females are taller than 5 feet.
B3. What percent of female adult heights are between 60 inches and 72
inches?
Using a single page Standard Normal Probability Distribution Table
starting at the mean and extending to z.
z = – 1.83 has a probability of 0.4664.
z = 2.17 has a probability of 0.4850.
0.4850 + 0.4664 = 0.9514 ≈ 95.1%
Using a two page Standard Normal Probability Distribution Table starting
at z and extending to the left.
z = 2.17 has a probability of 0.9850
z = – 1.83 has a probability of 0.0336
0.9850 – 0.0336 = 0.9514 ≈ 95.1%
The percent of female adult heights between 60 inches and 72 inches
is about 95.1%
C1. The bottom 8%
Using a single page Standard Normal Probability Distribution Table
starting at the mean and extending to z.
Look up 0.4200 in the table.
x = 65.5 + z(3)
x = 65.5 + (–1.41)(3)
x = 61.27
Using a two page Standard Normal Probability Distribution Table starting
at z and extending to the left.
Look up 0.0800 in the table.
x = 65.5 + z(3)
x = 65.5 + (–1.41)(3)
x = 61.27
The shortest female height for leather pants is approximately 5 ft 1 in.
C2. The upper 6%
Using a single page Standard Normal Probability Distribution Table starting
at the mean and extending to z.
Look up 0.4400 in the table.
x = 65.5 + z(3)
x = 65.5 + (1.555)(3)
x = 70.165
Using a two page Standard Normal Probability Distribution Table starting
at z and extending to the left.
Look up 0.9400 in the table.
x = 65.5 + z(3)
x = 65.5 + (1.555)(3)
x = 70.165
The tallest female height for leather pants is approximately 5 ft 10 in.
II. The male division
Male Data in inches
68
67.7
70.5
66.3
65.5
65.1
66.7
69.3
68.1
65.3
67.5
67.7
72.5
65.5
70.2
67
65.4
72
67.4
73.8
71.2
73.2
71.8
66.5
67.7
62.5
65.1
66.1
73.5
77.2
67.2
68.6
A. Using the calculator to find the sample mean and standard deviation (rounded
to hundredths) :
1. Mean height of an adult male is 68.5 inches.
2. Standard deviation of an adult male is 3.25 inches.
B1. What percent of male adults are shorter than 6 feet?
Using a single page Standard Normal Probability Distribution Table starting
at the mean and extending to z.
z = 1.08 has a probability of 0.3599.
0.5000 + 0.3599 = 0.8599 ≈ 86.0%
Using a two page Standard Normal Probability Distribution Table starting at
z and extending to the left.
z = 1.08 has a probability of 0.8599 ≈ 86.0%
The percent of male adults shorter than 6 feet is about 86%.
B2. What percent of male adults are shorter than 5 feet?
Using a single page Standard Normal Probability Distribution Table starting
at the mean and extending to z.
z = – 2.62 has a probability of 0.4956.
0.5000 – 0.4956 = 0.0044 ≈ 0.4%
Using a two page Standard Normal Probability Distribution Table starting
at z and extending to the left.
z = – 2.62 has a probability of 0.0044 ≈ 0.4%
The percent of male adults shorter than 5 feet is about 0.4%.
B3. What percent of male adult heights are between 60 inches and 72 inches?
Using a single page Standard Normal Probability Distribution Table starting
at the mean and extending to z.
z = – 2.62 has a probability of 0.4956.
z = 1.08 has a probability of 0.3599.
0.4956 + 0.3599 = 0.8555 ≈ 85.6%
Using a two page Standard Normal Probability Distribution Table starting at
z and extending to the left.
z = 1.08 has a probability of 0.8599.
z = – 2.62 has a probability of 0.0044
0.8599 – 0.0044 = 0.8555 ≈ 85.6%
The percent of male adult heights between 60 inches and 72 inches is about
85.6%.
C1. The bottom 9%
Using a single page Standard Normal Probability Distribution Table
starting at the mean and extending to z.
Look up 0.4100 in the table.
x = 68.5 + z(3.25)
x = 68.5 + (–1.34)(3.25)
x = 64.145
Using a two page Standard Normal Probability Distribution Table starting
at z and extending to the left.
Look up 0.0900 in the table.
x = 68.5 + z(3.25)
x = 68.5 + (–1.34)(3.25)
x = 64.145
The shortest male height for leather pants is approximately 5 feet 4 inches.
C2. The upper 7%
Using a single page Standard Normal Probability Distribution Table
starting at the mean and extending to z.
Look up 0.4300 in the table.
x = 68.5 + z(3.25)
x = 68.5 + (1.48)(3.25)
x = 73.31
Using a two page Standard Normal Probability Distribution Table starting
at z and extending to the left.
Look up 0.9300 in the table.
x = 68.5 + z(3.25)
x = 68.5 + (1.48)(3.25)
x = 73.31
The tallest male height for leather pants is approximately 6 feet 1 inch.