Sept. 9 Statistic for the day:
One in five people worldwide lives on
less than one dollar per day.
Smoothing the histogram:
The Normal Curve (Chapter 8)
A histogram tends to be rough. To replace it with a bell
shaped curve:
Center the bell at the mean.
The middle 95% of the bell should be 4 standard deviations.
Assignment: Exercises 2, 3, 8, 10, 16 from
Chapter 8 (pp. 156156-160)
This makes systematic, accurate predictions possible,
provided the bell shape is appropriate for the underlying
population.
Histogram of HandSpan, with Normal Curve
Histogram of HandSpan, with Normal Curve
Frequency
20
10
0
10
0
15
20
25
15
HandSpan
25
Mean = 20.86
Standard deviation = 1.927
Histogram of Height, with Normal Curve
Research Question 1: If I built my doors
75 inches (6 feet 3 inches) high, what
percent of the people would have to
duck?
30
20
10
0
60
20
HandSpan
Mean = 20.86
Standard deviation = 1.927
Frequency
Frequency
20
70
Height
Mean = 68 inches or 5 feet 8 inches
Standard deviation = 4 inches
80
Research Question 2: How high should
I build my doorways so that 99% of the
people will not have to duck?
1
Histogram of Height, with Normal Curve
Z-Scores: Measurement in Standard
Deviations
Frequency
30
Given the mean (68), the standard deviation (4), and a
value (height say 75) compute
20
10
Z=
0
60
70
80
75 − mean 75 − 68
=
= 1.75
SD
4
Height
Question 1
(x=75)
Question 2
(x=??)
This says that 75 is 1.75 standard deviations
above the mean.
Q1: The value of x is 75; find the amount of distribution above it.
Q2: Find the value of x so that 99% of the distribution is below it.
Compute your Z-score.
Compare Heights of Females and Males
Stat 100 students Sp01
1. How many standard deviations are you
above or below the mean.
Height
Use:
Mean = 68 inches
Standard deviation = 4 inches
80
70
60
2. Now use the table from the book (p. 157) to
determine what percentile you are.
Female
Male
Sex
Assume male heights have a normal distribution with
mean 70 and st dev 3. Assume female heights have a
normal distribution with mean 64 and st dev 3.
Answer to Question 1: What percent of people would
have to duck if I built my doors 75 inches high?
What is your Z-Score within your sex?
From the standard normal table in the book: .96 or
96% of the distribution is below 1.75. Hence, .04
or 4% is above 1.75.
What is your percentile within your sex?
Recall: 75 has a Z-score of 1.75
So 4% of the distribution is above 75 inches.
2
Question 2: What is the value so that 99% of the
distribution is below it? (called the 99th percentile.)
Histogram of Height, with Normal Curve
Frequency
30
1. Look up the Z-score that corresponds to the 99th
percentile. From the table: Z = 2.33.
20
4% in here
2. Now convert it over to inches:
10
2.33 =
0
60
70
75
80
h99 = 68 + 2.33(4) = 77.3
Height
Question 1
(x=75)
The value at x is 75; find the amount of distribution above
it. Convert 75 to Z = 1.75 and use Table 8.1 on p. 157.
h99 − 68
4
Therefore, 99% of the distribution is shorter than 77.3
inches (6 foot 5.3 inches) and that’s how high the door
should be built.
Histogram of Height, with Normal Curve
Answer these questions:
Frequency
30
Research Question 1: What percent of people are less
than six feet (72 inches) tall?
20
10
Research Question 2: What is the first quartile of
heights?
99% in here
0
60
70
80
Height
Question 2
77.3 inches is the 99th percentile
To find the value so that 99% of the distribution is below it: Look
up the Z-score for the 99th percentile and convert it back to inches.
Z-Scores: Measurement in Standard
Deviations
Given the mean (68), the standard deviation (4), and a
value (say 72), compute
Z = (72(72-mean) / SD = (72(72-68) / 4 = 1
This says that 72 is 1 standard deviation above the
mean.
What proportion of heights are below ZZ-score=1 ?
(Assume that adults’ heights are normally distributed
with mean 68 inches and standard deviation 4 inches.)
Answer to Question 2: What is the first quartile of heights?
Translation: “First quartile” means 25th percentile, which
means .25 are below that height.
From p. 157: Find the z-score corresponding to the 25th
percentile.
Now convert this z-score into a height:
Z − score =
h − 68
4
h = 68 + 4( Z − score)
3
Shaquille O’Neal is 7 feet 1 inch or
85 inches tall. How many people in
the country are taller?
There are roughly 295 million people in US.
About 49% are over the age of 20 (Census Bureau).
That is 144.5 million.
Hence, there should be roughly
.000011 times 144.5 million
or 1500 people taller than Shaquille O’Neal.
We will assume that heights are normally distributed
with mean 68 inches and standard deviation 4 inches.
O’Neal’s Z-score is Z = (85-68)/4 = 4.25. In other words
O’Neal is 4.25 standard deviations above the mean(!)
Note: This is an extremely rough calculation, since the
normal distribution approximation is less accurate at the
extremes. Also, cutting off at age 20 might miss some
tall teens!
There is only 0.000011 of the normal distribution above
4.25 standard deviations.
Page 157
Suppose someone claims to have
tossed a fair coin 100 times and got
70 heads. Would you believe them?
„ We need to know what the distribution of the number
of heads in 100 tosses looks like for a fair coin.
„ We need the mean and standard deviation for this
distribution.
3.
Determine the standard score in a normal distribution that has
the following percentage below it:
a. 25%
b. 75%
c. 45%
d. 98%
4. Determine the standard score in a normal distribution that has
the following percentage above it:
a. 2%
b. 50%
c. 75%
d. 10%
Toss a coin 100 times
Repeat 500 times and form a histogram
90
80
70
Frequency
1. Determine the percentage of a normal distribution
falling below each of the following standard scores:
a. -1.00
b. 1.96
c. 0.84
2. Determine the percentage of a normal distribution
falling above each of the following standard scores:
a. 1.28
b. -0.25
c. 2.33
Page 158
60
50
40
30
20
10
0
35
45
55
Number of heads
65
1. What is the mean?
2. What is the standard deviation?
3. Let’s suppose the smooth version is normal.
4
So the distribution of the number of
heads in 100 tosses of a fair coin is:
„
„
„
„
„
„
„
„
Roughly normal, mean about 50, SD about 5
What is the ZZ-score of 70?
Ans:
Ans: 4
What is the percentile?
Ans:
Ans: .999968 or 99.9968%
Now do you believe them?
NO
Weighted coin is a BETTER explanation
5