Download Section 11.3: Normal Distribution and Empirical Rule

Advanced Algebra
DATA ANALYSIS AND STATISTICS
Name _______________________________________
Period ______ Date ___________________________
Section 11.3: Normal Distribution and Empirical Rule
In a normal distribution, what percent of the values lie:
1. below the mean?
2. above the mean?
3. within one standard deviation of the mean?
4. within two standard deviations of the mean?
5. within three standard deviations of the mean?
6.
In Advanced Algebra the mean for the last chapter test was 80 points with a standard deviation of 7
points. Assume the test scores for the subject test are normally distributed.
a) Create a bell curve for the test scores (use the one provided below.) Make sure to clearly label your
horizontal axis showing appropriate values for the mean as well as ± 1 standard deviation, ± 2
standard deviations, and ± 3 standard deviations.
Use your graph to answer each of the following questions:
b) What percent of the students who took the test scored below 73 points? Justify your answer.
c) There are 400 students enrolled in Advanced Algebra. How many of the students scored below 73
points? Show all your work.
d) What percent of the students who took the test scored above 94 points? Justify your answer.
e) There are 400 students enrolled in Advanced Algebra. How many of the students scored above 94
points? Show all your work.
f) What percent of the students who took the test scored between 59 and 66 points? Justify your
answer.
g) There are 400 students enrolled in Advanced Algebra. How many of the students scored between
59 and 66 points? Show all your work.
SEE OTHER SIDE
7. For a sample of studio apartments in San Francisco, 68% of the monthly rents are between $1145 and
$1375. Assume the monthly rents for the subject apartments are normally distributed.
a) Create a bell curve for monthly rents (use the one provided below.) Make sure to clearly label your
horizontal axis showing appropriate values for the mean as well as ± 1 standard deviation, ± 2
standard deviations, and ± 3 standard deviations.
b) Find the mean and standard deviation of the distribution.
c) Find the probability of finding a studio apartment for less than $915 a month.
d) Find the probability of finding a studio apartment for more than $1490 a month.
e) Find the probability of finding a studio apartment with a month rent between $1030 and $1375.
8. Use the data set of pulse rates below to answer the following questions.
68 105 95 80 90 100 75 70 84 98 102 70
65 88 90 75 78 94 110 112 95 80 76 108
a) Use your calculator to find measures of spread and center.
Median = _________ IQR = _________
Mean = _________ Standard Deviation = _________
b) Construct a frequency distribution table and a histogram of the data.
Pulse Rates
Frequency
60 to < 70
70 to < 80
80 to < 90
90 to < 100
100 to < 110
110 to < 120
c) Construct a Box and Whisker Plot of the data. Clearly label the five-number summary.
Advanced Algebra
ANSWER KEY
DATA ANALYSIS AND STATISTICS
Normal Distribution and Empirical Rule
In a normal distribution, what percent of the values lie:
1. below the mean? 50%
2. above the mean? 50%
3. within one standard deviation of the mean? 68%
4. within two standard deviations of the mean? 95%
5. within three standard deviations of the mean? 99.7%
6.
In an Advanced Algebra class the mean for the last chapter test was 80 points with a standard
deviation of 7 points. Assume the test scores for the subject test are normally distributed.
a)
59
66
73
80
87
94
101
b) What percent of the students who took the test scored below 73 points? Justify your
answer.
73 = 80 − 7, so 73 is one standard deviation below the mean.
Therefore 𝑷(𝐬𝐜𝐨𝐫𝐞 < 𝟕𝟑) = 𝟏𝟔%
c) There are 400 students enrolled in Advanced Algebra. How many of the students scored
below 73 points? Show all your work.
𝟏𝟔% of 400 is 64, so 64 students scored below 73 points
d) What percent of the students who took the test scored above 94 points? Justify your
answer.
94 = 80 + 2×7, so 94 is two standard deviations above the mean.
Therefore 𝑷(𝐬𝐜𝐨𝐫𝐞 > 𝟗𝟒) = 𝟐. 𝟓%
e) There are 400 students enrolled in Advanced Algebra. How many of the students scored
above 94 points? Show all your work.
𝟐. 𝟓% of 400 is 10, so 10 students scored above 94 points
f) What percent of the students who took the test scored between 59 and 66 points? Justify
your answer.
59 = 80 – 3×7 and 66 = 80 − 2×7, so the data is contained between three and two standard
deviations below the mean
Therefore 𝑷(𝟓𝟗 < 𝐬𝐜𝐨𝐫𝐞 < 𝟔𝟔) = 𝟐. 𝟑𝟓%
g) There are 400 students enrolled in Advanced Algebra. How many of the students scored
between 59 and 66 points? Show all your work.
𝟐. 𝟑𝟓% of 400 is 9.4, so about 9 students scored between 59 and 66 points
7. For a sample of studio apartments in San Francisco, 68% of the monthly rents are between $1145 and
$1375. Assume the monthly rents for the subject apartments are normally distributed.
a) Create a bell curve for monthly rents.
915
1030
1145
1260
1375
1490
1605
b) Find the mean and standard deviation of the distribution.
Mean = 1260; Standard deviation = 115
c) Find the probability of finding a studio apartment for less than $915 a month.
0.15%
d) Find the probability of finding a studio apartment for more than $1490 a month.
2.35%
e) Find the probability of finding a studio apartment with a month rent between $1030 and $1375.
81.5%
8. Use the data set of pulse rates below to answer the following questions.
68 105 95 80 90 100 75 70 84 98 102 70
65 88 90 75 78 94 110 112 95 80 76 108
a) Use your calculator to find measures of spread and center.
Median = 89 IQR = 23.5
Mean = 87.833 Standard Deviation = 13.894
b) Construct a frequency distribution table and a histogram of the data.
Frequency
60 to < 70
2
70 to < 80
6
80 to < 90
4
90 to < 100
6
100 to < 110
4
110 to < 120
2
Pulse Rates
Frequency
Pulse Rates
7
6
5
4
3
2
1
0
60 to 69
70 to 79
80 to 89
90 to 99 100 to 109 110 to 119
Pulse Rate per minute
c) Construct a Box and Whisker Plot of the data. Clearly label the five-number summary