Download Lecture 5

Sept. 2
Reading Assignment:
Finish Chapter 8; skim through Chapter 7 again
Try exercises 15 and 27 on pages 159–161
20
What is the best estimate below of the s.d. for these data?
10
5
0
Frequency
15
(A)  10
(B)  20
(C)  30
(D)  40
(E)  50
40
60
80
100
120
140
How much does this cow weigh?
(The man in the
picture weighs
165 pounds.)
Histogram of Height, with Normal Curve
Frequency
30
20
10
0
60
70
Height
Mean = 68 inches or 5 feet 8 inches
Standard deviation = 4 inches
80
Assume human heights are normally distributed with a
mean of µ=68 inches and a standard deviation of σ=4
inches.
Research Question 1: If I built my doors 75 inches (6
feet 3 inches) high, what percent of the people would
have to duck?
Research Question 2: How high should I build my
doorways so that 99% of the people will not have to
duck?
Z-Scores: Measurement in Standard
Deviations
Given the mean (68 inches), the standard deviation (4
inches), and a value (75 inches) compute
75 − mean 75 − 68
Z=
=
= 1.75
SD
4
This says that 75 is 1.75 standard deviations
above the mean.
Z-Scores: Measurement in Standard
Deviations
Given the mean (68 inches), the standard deviation (4
inches), and a value (60 inches) compute
Z = ??
(A)  Z = –2.0
(B)  Z = –1.0
(C)  Z = 0.0
(D)  Z = 1.0
(E)  Z = 2.0
Height=75 inches gives z-score=+1.75
What percent?
-3
-2
-1
0
1
2
3
Use table on p. 175 to find percent to the left of Z=+1.75.
Here is what p. 157 tells us:
Should be p. 175
Z=+1.75
96%
-3
-2
-1
0
1
2
3
What percent of people would have to duck if I built my doors
75 inches high? Ans: 4%
Question 2: What is the value so that 99% of the
distribution is below it? (called the 99th percentile.)
1.  Look up the Z-score that corresponds to the 99th
percentile. From the table: Z = 2.33.
2.  Now convert it to inches:
h99 − 68
2.33 =
4
h99 = 68 + 2.33(4) = 77.3
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.
What Z?
99%
p. 175 tells us that the Z-score at the 99th percentile is 2.33.
In inches, 2.33 standard deviations above the mean is 77.3.
Assume male heights have a normal distribution with
mean 70 inches and st dev 3 inches. Assume female
heights have a normal distribution with mean 64 inches
and st dev 3 inches.
What is your Z-Score within your sex?
What is your percentile within your sex?
Shaquille O’Neal is 7 feet 1 inch or
85 inches tall. How many people in
the country are taller?
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(!)
There is only 0.000011 of the normal distribution above
4.25 standard deviations.
There are roughly 317 million people in US.
About 49% are over the age of 20 (Census Bureau).
That is about 155 million.
Hence, there should be roughly
.000011 times 155 million
or 1705 people taller than Shaquille O’Neal.
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!
Table 8.1 practice (page 175)
Match the percentages of a
normal distribution with the
following standardized (z-)
scores:
1.  below z = −1.00
2.  below z = 1.96
3.  above z = 0.84
4.  above z = 0
5.  below z = –0.50
(A)  20%
(B)  97.5%
(C)  50%
(D)  16%
(E)  31%
Table 8.1 practice (page 175)
Match the percentages of a
normal distribution with the
following standardized (z-)
scores:
1.  below z = −1.00
2.  below z = 1.96
3.  above z = 0.84
4.  above z = 0
5.  below z = –0.50
(A)  20%
(B)  97.5%
(C)  50%
(D)  16%
(E)  31%
Table 8.1 practice (page 175)
Match the percentages of a
normal distribution with the
following standardized (z-)
scores:
1.  below z = −1.00
2.  below z = 1.96
3.  above z = 0.84
4.  above z = 0
5.  below z = –0.50
(A)  20%
(B)  97.5%
(C)  50%
(D)  16%
(E)  31%
Table 8.1 practice (page 175)
Match the percentages of a
normal distribution with the
following standardized (z-)
scores:
1.  below z = −1.00
2.  below z = 1.96
3.  above z = 0.84
4.  above z = 0
5.  below z = –0.50
(A)  20%
(B)  97.5%
(C)  50%
(D)  16%
(E)  31%
Table 8.1 practice (page 175)
Match the percentages of a
normal distribution with the
following standardized (z-)
scores:
1.  below z = −1.00
2.  below z = 1.96
3.  above z = 0.84
4.  above z = 0
5.  below z = –0.50
(A)  20%
(B)  97.5%
(C)  50%
(D)  16%
(E)  31%
Table 8.1 practice (page 175)
Match the percentages of a
normal distribution with the
following standardized (z-)
scores:
1.  below z = −1.00
2.  below z = 1.96
3.  above z = 0.84
4.  above z = 0
5.  below z = –0.50
(A)  20%
(B)  97.5%
(C)  50%
(D)  16%
(E)  31%
Table 8.1 practice (page 175)
Match the standardized
(z-)scores with the following
percentiles of a standard
normal distribution:
1.  25th %ile
2.  75th %ile
3.  42nd %ile
4.  98th %ile
5.  10th %ile
(A)  –0.67
(B)  0.67
(C)  2.05
(D)  0.20
(E)  –1.28
Table 8.1 practice (page 175)
Match the standardized
(z-)scores with the following
percentiles of a standard
normal distribution:
1.  25th %ile
2.  75th %ile
3.  42nd %ile
4.  98th %ile
5.  10th %ile
(A)  –0.67
(B)  0.67
(C)  2.05
(D)  0.20
(E)  –1.28
Table 8.1 practice (page 175)
Match the standardized
(z-)scores with the following
percentiles of a standard
normal distribution:
1.  25th %ile
2.  75th %ile
3.  42nd %ile
4.  98th %ile
5.  10th %ile
(A)  –0.67
(B)  0.67
(C)  2.05
(D)  0.20
(E)  –1.28
Table 8.1 practice (page 175)
Match the standardized
(z-)scores with the following
percentiles of a standard
normal distribution:
1.  25th %ile
2.  75th %ile
3.  42nd %ile
4.  98th %ile
5.  10th %ile
(A)  –0.67
(B)  0.67
(C)  2.05
(D)  0.20
(E)  –1.28
Table 8.1 practice (page 175)
Match the standardized
(z-)scores with the following
percentiles of a standard
normal distribution:
1.  25th %ile
2.  75th %ile
3.  42nd %ile
4.  98th %ile
5.  10th %ile
(A)  –0.67
(B)  0.67
(C)  2.05
(D)  0.20
(E)  –1.28
Table 8.1 practice (page 175)
Match the standardized
(z-)scores with the following
percentiles of a standard
normal distribution:
1.  25th %ile
2.  75th %ile
3.  42nd %ile
4.  98th %ile
5.  10th %ile
(A)  –0.67
(B)  0.67
(C)  2.05
(D)  0.20
(E)  –1.28