Download Normal Distribution & Binomial Probabilities

Aim: How do we apply the characteristics of
normal distribution?
Do Now:
10 coins tossed 100 times result in the
following table. Draw a histogram based
on the table and determine the mean, x.
# of heads
0 1 2 3 4 5 6 7 8 9 10
Frequency
1
2
4 11 20 24 20 11 4
25
20
Frequency
15
10
5
0
0 1 2 3 4 5 6 7 8 9 10
Aim: Normal Distribution
# of Heads
Course: Alg. 2 & Trig.
2
1
Normal Curve – the ‘Bell Curve’
mean x
also
mode &
median
symmetrical
The most prominent probability
distribution in statistics.
50
40
30
20
10
Aim:
0 Normal Distribution
0
1
2
3
Course: Alg. 2 & Trig.
4
5
6
7
8
9 10
Normal Distribution
99.5% of data values
95% of data values
68% of data values
13.5% 34% 34% 13.5%
x  3 x  2 x  
x
x 
x  2 x  3
68% of data lie within 1 standard deviation of mean.
95% of data within 2 standard deviations of mean.
Aim: Normal Distribution
Course: Alg. 2 & Trig.
99.5% of data within 3 standard deviations of mean.
Percentile
99.5% of data values
95% of data values
68% of data values
13.5% 34% 34% 13.5%
x  3 x  2 x  
2.5
16
x
50
x 
84
x  2 x  3
97.5
percentile of a score or a measure indicates
what percent of the total frequency scored at
Normal Distribution
Course: Alg. 2 & Trig.
or belowAim:that
measure.
Model Problem
In a normal distribution, the mean height of
10-year-old children is 138 centimeters
and the standard deviation is 5
centimeters. Find the heights that are
a) exactly one standard deviation above and
x    138  5  143
below the mean
x    138  5  133
b) two standard deviations above and below
the mean
x  2  138  2  5   148
x  2  138  2  5   128
x  138
 5
X  2Alg.
 2 & Trig.
X  Course:
138
128 133
143 148
X  2 X  
Aim: Normal Distribution
10-year-old Model Problem
In a normal distribution, the mean height of 10-yearold children is 138 centimeters and the standard
deviation is 5 centimeters.
95%
68%
13.5% 34% 34% 13.5%
X  2 X  
128 133
2.5 16
138
X   X  2
143 148
50 84 97.5
Of the children:
68% are between 133 and 143 centimeters tall
95% are between 128 and 148 centimeters tall
34% are between 138 and 142 centimeters tall
Aim: Normal Distribution
Course: Alg. 2 & Trig.
10-year-old Model Problem
In a normal distribution, the mean height of 10-yearold children is 138 centimeters and the standard
deviation is 5 centimeters.
13.5% 34% 34% 13.5%
X  2 X  
128 133
2.5 16
138
X   X  2
143 148
50 84 97.5
A ten-year-old who is 133 cm. tall is at the
16th percentile; 16% are shorter, 84% taller
Heights that would occur less than 5% of
the time: heights of less than 128 cm. or
more than 148 cm.
Aim: Normal Distribution
Course: Alg. 2 & Trig.
2pt. Regents Question
Assume that the ages of first-year college
students are normally distributed with a
mean of 19 years and standard deviation of
1 year.
To the nearest integer, find the percentage of
first-year college students who are between
the ages of 18 years and 20 years inclusive.
To the nearest integer, find the percentage of
first-year college students who are 20 years
or older.
Aim: Normal Distribution
Course: Alg. 2 & Trig.
Model Problem
Scores on the Preliminary Scholastic Aptitude
Test (PSAT) range from 20 to 80. For a
certain population of students, the mean is
52 and the standard deviation is 9.
a) A score at the 65th percentile might be
1) 49
2) 56
3) 64
4) 65
b) Which of the following scores can be
expected to occur less than 3% of the time?
1) 39
2) 47
3) 65
4) 71
13.5% 34% 34% 13.5%
X  2 X  
52
34 43
2.5 16 50
Aim: Normal Distribution
X 
X  2
61 70
84 97.5
Course: Alg. 2 & Trig.
Model Problem
In the diagram, the shaded area represents
approximately 68% of the scores in a
normal distribution. If the scores range
from 12 to 40 in this interval, find the
standard deviation.
 x      x     40  12
2  28
  14
13.5% 34% 34% 13.5%
X  2 X  
Aim: Normal Distribution
X
2.5
50
12
16
X 
X  2
40 Course: Alg. 2 & Trig.
84 97.5
4pt. Regents Question
Twenty high school students took an
examination and received the following
scores:
70, 60, 75, 68, 85, 86, 78, 72, 82, 88, 88, 73,
74, 79, 86, 82, 90, 92, 93, 73
Determine what percent of the student
scored within one standard deviation of the
mean. Do the results of the examination
approximate a normal distribution? Justify
your answer.
Aim: Normal Distribution
Course: Alg. 2 & Trig.
Model Problem
In 2000, over 1.2 million students across the
country took college entrance exams. The
average score on the verbal section showed
no improvement over the average scores of
the previous 4 years. The average score on
the mathematics section was 3 points higher
than the previous year’s average.
Section
Mean
Math
Verbal
505
514
Standard
Deviation
111
113
What is the probability that a student’s
verbal score is from 401 to 514?
Aim: Normal Distribution
Course: Alg. 2 & Trig.
Model Problem
In 2000, over 1.2 million students across the
country took college entrance exams. The
average score on the verbal section showed
no improvement over the average scores of
the previous 4 years. The average score on
the mathematics section was 3 points higher
than the previous year’s average.
Section
Mean
Math
Verbal
505
514
Standard
Deviation
111
113
What is the probability that a student’s
math score is greater than 727?
Aim: Normal Distribution
Course: Alg. 2 & Trig.
Model Problem
In 2000, over 1.2 million students across the
country took college entrance exams. The average
score on the verbal section showed no improvement
over the average scores of the previous 4 years. The
average score on the mathematics section was 3
points higher than the previous year’s average.
Section
Mean
Math
Verbal
505
514
Standard
Deviation
111
113
Both Susanna’s math and verbal scores were
more than one standard deviation above the
mean, but less than 2 standard deviations
above the mean. What are the lower and
upper limits
of Distribution
Susanna’s combined
score?
Aim: Normal
Course: Alg.
2 & Trig.