Download Algebra 2 23.2 Notes 23.2 (Day One) Normal Distributions Date

Algebra 2
23.2 Notes
23.2 (Day One) Normal Distributions
Date: ___________
Finding Areas Under a Normal Curve
Learning Target A: I can use the 68-95-99.7 rule to analyze data that is normally distributed.
Normal Distribution: in a Normal Distribution, when you draw a smooth curve close to or
through the tops of the rectangles of a histogram, you get a _________-shaped curve like the one
shown. The curve is symmetrical about the ____________.
**In a normal distribution, the mean and median are _________________.
The 68-95-99.7 Rule for Normal Distributions
For a normal distribution:



About 68% of the data are within ________
standard deviation of the mean.
About 95% of the data are within ________
standard deviations of the mean.
About 99.7% of the data are within ___________
standard deviations of the mean.
We can split these percentages up into intervals of single standard deviations above or below the mean.
Using the standard normal curve, find each of the following:
1. The percent of data between one standard deviation below the mean and two standard deviations
above the mean.
2. The percent of data more than two standard deviations above the mean.
3. The percent of data that falls below one standard deviation below the mean.
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Algebra 2
23.2 Notes
Use the 68-95-99.7 Rule to complete each problem. It is helpful to draw a bell-shaped curve with
the given mean and standard deviations.
Example 1. Suppose the heights (in inches) of men (ages 20-29) years old in the United States are
normally distributed with a mean of 69.3 inches and a standard deviation of 2.92 inches. Find each of
the following.
A) Draw a bell-shaped curve to model the situation.
B) The percent of men who are between 63.46 inches and 75.14 inches tall.
C) The percent of men who are taller than 72.22 inches
D) The percent of men who are between 60.54 and 78.06 inches tall.
E) The percent of men who are shorter than 60.54 inches.
Example 2. In 1992-1993, the annual salaries of the 43 state lieutenant governors in the US had a
normal distribution with 𝜇 = $57,000 and 𝜎 = $22,000.
A) Draw a bell-shaped curve to model the situation.
B) Within what range do about 95% of lieutenant governors’ salaries fall?
C) About what percent of the lieutenant governors have an annual salary between $13,000 and
$79,000?
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Algebra 2
23.2 Notes
Example 3. At Hamilton High, the heights of female students are approximately normally distributed
with 𝜇 = 64.5 𝑖𝑛𝑐ℎ𝑒𝑠 and 𝜎 = 2.5 𝑖𝑛𝑐ℎ𝑒𝑠.
A) Within what range of heights do 95% of the female students fall?
B) About what percent of female students have heights between 62 and 67 inches?
C) If Hamilton has 2100 female students, how many would you expect to be over 6 feet tall?
Example 4. Sometimes people who make airplane reservations do not show up for the flight. Suppose
an airline finds that for a certain route, the number of “no-shows” is normally distributed with a mean of
20 and standard deviation of 5.
A) For 95% of the flights, the number of “no-shows” is between what two numbers?
B) For 84% of flights, the number of “no-shows” is above what number?
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Algebra 2
23.2 Notes
23.2 (Day Two) Normal Distributions
Date: ___________
Explore. Who’s the Better Student?
Fantastic Public Schools gives a math test to all its students every year. It is given in 3rd grade, 7th
grade, and 10th grade. It is called the Stanford Achievement Test. The scores from one year are
given in the table below
Grade
Level
3
7
10
Mean Number of
Correct Answers
102.4
128.7
146.0
Standard
Deviation
16.3
20.0
18.1
Gina is in third grade and scored 113 on the math test. Michelle is in tenth grade and scored 157
on the same test.
a) Is it fair to compare their scores directly and conclude that Michelle is a better math student?
b) Find the difference between each girl’s score and the mean for her grade level.
c) Is it better to be above or below the mean?
d) Divide the difference by the standard deviation for each grade level. This tells how many standard
deviations from the mean each girl scored. How many standard deviations from the mean was each
girl’s score?
e) Based on this information, which girl is the better math student?
Explain 2. Using a Standard Normal Distribution
Learning Target B. I can find and use z-scores to analyze data that is normally distributed.
Standard Normal Distribution: has a mean of 0 and a standard deviation of 1.
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Algebra 2
23.2 Notes
Standardizing Data Using Z-Scores: A data value x from a normal distribution with mean 𝜇 and
standard deviation 𝜎 can be standardized by finding its z-score.
𝑧=
𝑥−𝜇
𝜎
** A z-score tells how many standard deviations a data value is above or below the mean. The area under the
standard normal curve represents the percent of data that fall under the data value associated with the z-score.
Sketch a standard normal curve and shade the region under the curve for the values of z less
than the given score.
a) 𝑧 = −3.2
b) 𝑧 = 2.1
Standard Normal Table- Gives the percentage of data that falls _________________ a data value given its
z-score.
z
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
-3
.0013
.0010
.0007
.0005
.0003
.0002
.0002
.0001
.0001
.0000+
-2
.0228
.0179
.0139
.0107
.0082
.0062
.0047
.0035
.0026
.0019
-1
.1587
.1357
.1151
.0968
.0808
.0668
.0548
.0446
.0359
.0287
-0
.5000
.4602
.4207
.3821
.3446
.3085
.2743
.2420
.2119
.1841
0
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
1
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
2
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
3
.9987
.9990
.9993
.9995
.9997
.9998
.9998
.9999
.9999
1.0000-
Example 1. Give the z-score for each of the following percentages.
A) 34.46%
B) 94%
D) 0.05%
Example 2. Find the percentage of data values that fall under the data value with the given z-score.
A) 𝑧 = 0.5
B) 𝑧 = 1.8
C) 𝑧 = −0.3
Example 3. Find the percent of data values that fall between the values with the given z-scores
A) 𝑧 = 0.9 and 𝑧 = 1.7
B) 𝑧 = 3.0 and 𝑧 = 2.9
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Algebra 2
23.2 Notes
Example 4. Suppose the heights (in inches) of women (ages 20-29) in the United States are normally
distributed with a mean of 64.1 inches and standard deviation of 2.75 inches.
A) Find the percent of women who are less than or equal to 65 inches tall.
B) Find the percent of women who are at least 65 inches tall.
C) Find the percent of women who are between 60 and 63 inches tall.
D) Find the percent of women who are no more than 66 inches tall.
E) Find the percent of women who are less than or equal to 61.6 inches tall.
Example 5. The mean score on the last test given in a history class was 78.1 with a standard deviation
of 10.9. The scores were normally distributed.
A) Jerome scored an 80. What percent of the students did Jerome perform better than?
B) Brandt had a score higher than 86% of students. What was his score?
C) What percent of students scored between 65 and 72?
D) If Michael scored a 91, what percent of students scored higher than he did?
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