7.6: Normal Distributions
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Do these two normal curves have the same mean?
Do they have the same standard deviation?
Standard Normal Distribution
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Using the Table of Standard (z) Curve Areas
P(z<-1.62)=
P(z<2.31)=
P(z>2.31)=
P(z<zk)=.02
P(z>zk)=.05
Finding Probabilities for Other Normal Curves
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1. Data on the length of time to complete registration for classes using an on-line registration
system suggest that the distribution of the variable x=time to register for students at a particular
university can be well approximated by a normal distribution with mean 12 minutes and standard
deviation of 2 minutes.
a) What is the probability that it will take a randomly selected student less than 9 minutes to
compete registration?
b) What is the probability that it will take a randomly selected student more than 13 minutes to
complete registration?
c) What is the probability that it will take a randomly selected student between 7 and 15 minutes
to complete registration?
d) Because some students do not log off properly, the university would like to log off student
automatically after some time has elapsed. It is decided to select this time so that only 1% of
students will be automatically logged off while still trying to register. What time should the
automatic log off be set at?
Calculator method!
2nd -> DISTR
Normalcdf(lower, upper, mean, standard deviation)
invNorm(probability, mean, standard deviation)
2. Determine the following areas under the standard normal (z) curve.
a)
The area under the z curve to the left of 1.56
b)
The area under the z curve to the left of –2.13
c)
The area under the z curve to the right of 0.88
d)
The area under the z curve to the right of –1.23
e)
The area under the z curve between –2 and 3
f)
The area under the z curve between –2.4 and –1.4
3. Let z denote a random variable having a standard normal distribution. Determine each of the
following probabilities.
a)
P(z < 1.36)
b)
P(z < - 2.05)
c)
P(z > - 1.15)
d)
P(- 1.10 < z < 2.54)
4. The time that it takes a randomly selected employee to perform a certain task is approximately
normally distributed with a mean value of 120 seconds and a standard deviation of 20 seconds. The
slowest 10% (that is, the 10% with the longest times) are to be given remedial training. What times
qualify for the remedial training?
5. The distribution of IQ scores is normal, with a mean of 100 and standard deviation of 15. One
way to become eligible for membership in Mensa, a person must demonstrate an IQ of 130. What
proportion of the population would qualify for membership in Mensa?
6. In poor countries, the growth of children can be an important indicator of general levels of
nutrition and health. Data from the article “The Osteological Paradox: Problems in Inferring
Prehistoric Health from Skeletal Sample” suggest that a reasonable model for the probability
distribution of the continuous numerical variable x = height of a randomly selected 5 year old child
is a normal distribution with a mean of µ = 100 cm and standard deviation σ = 6 cm.
a) What proportion of the heights is between 94 and 112 cm?
b) What is the probability that a randomly chosen child will be taller than 110 cm?
7.7 Checking for Normality and Normalizing Transformations
Some of the most frequently used statistical methods are valid only when x1, x2,…,xn has come from
a population distribution that at least is approximately normal. One way to see whether an
assumption of population normality is plausible is not construct a normal probability plot of the
data.
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Let’s construct a normal probability plot. The normal scores when n=10 are:
-1.539 -1.001 -.656 -.376 -.123 .123
.376
.656
1.001 1.539
Egg weights in grams for a sample of 10 eggs:
53.04 53.50 52.53 53.00 53.07 52.86 52.66 53.23 53.26 53.16
Sketch a scatter plot, pairing the smallest normal score with the smallest value in the sample:
Using the Correlation Coefficient to Assess Normality
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Values to Which r can be Compared to Check for Normality
N
Critical r
5
.832
10
.880
15
.911
20
.929
25
.941
30
.949
40
.960
50
.966
60
.971
70
.976
7. Check the following data set for normality by creating a scatter plot . (TI-83/84: Make a
STATPLOT, using the last graph type)
16.01
16.04
16.07
16.08
15.82
16.13
16.13
16.15
16.11
15.94
16.06
16.05
15.66
16.27
15.78
15.89
15.99
15.84
16.29
15.95
16.15
16.10
16.19
15.92
16.22
8. Given the following table of critical r values, data sets and correlations, which data set looks like
it has the least chance of being normal?
(A) n = 4096 ; r = 0.964
(B) n = 20 ; r = 0.98
(C) n = 22 ; r = 0.96
(D) n = 600 ; r = 0.9999
(E) n = 40 ; r = 0.961
9. Based on the normal probability plot below, does it appear that a normal probability model is
appropriate for this model? Explain.
10. The normal probability plots below have “confidence bands”. For a normal probability model to
be appropriate, almost all points should be within the confidence bands. For each of the normal
probability plots below, does it appear that a normal probability model is appropriate? Explain.
11. IQ scores for a random sample of people are shown below.
72
79
87
91
99
101
103
106
111
113
116
126
Make a normal probability plot and sketch it below.
Transforming Data to Achieve Normality
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7.8 Using the Normal Distribution to Approximate a Discrete Distribution
Suppose the probability distribution of a discrete random variable x is displayed in the histogram
below.
Approximately Normal Distribution:
Continuity Correction:
Normal Approximation to a Binomial Distribution
12. Premature babies are born before 37 weeks, and those born before 34 weeks are most at risk. A
study reported that 2% of births in the US occur before 34 weeks. Suppose that 1000 births are
randomly selected and that the number of these births that occurred prior to 34 weeks, x is to be
determined.
Can the distribution of x be approximated by a normal distribution?
Find the mean and standard deviation for the approximated normal distribution.
What is the probability that the number of babies in the sample of 1000 born prior to 34 weeks will
be between 10 and 25 (inclusive)?
13. Among 100 randomly selected bike purchases the expected value of mountain bikes is 70 and the
standard deviation is 5. What is the approximate probability that:
a) At most 74 are mountain bikes?
b) Strictly between 60 and 74 mountain bikes?
c) More than 80 are mountain bikes
d) Exactly 70 are mountain bikes?
14.
In China 63% of children ages 0-14 are boys. Suppose that 16 children ages 0-14 are
randomly selected and x is the number of these children that are boys. Find the following:
μ = ___________________
σ = ___________________
Can it be assumed that x has an approximately a normal distribution? Explain how you know.