Download 6.4 The Standard Normal Distribution

6.4 The Standard Normal Distribution
Upon completion of this activity and any associated homework, you should be
able to:
• Shade in an area under a normal curve that represents the probability of
a continuous random variable taking on an interval of values
• Use a graphing calculator to find probabilities for a continuous random
variable
• Use a graphing calculator to find data values given the probability of a
continuous random variable taking on an interval of values
1. The height X (in inches) of women in the United States is considered to
be normally distributed with a mean of 64 and standard deviation of 2.7.
This is denoted N (64, 2.7)
X represents:
N (64, 2.7) means:
(a) Use the 68 − 95 − 99.7 rule to shade the following probability density
curves for the normal distribution of heights of women.
i. 68% of the data is within one standard deviation of the mean, so
the area representing the probability that a randomly selected
woman is within one standard deviation of the mean is
.
ii. 95% of the data is within two standard deviations of the mean,
so the area representing the probability that a randomly selected
.
woman is within two standard deviation of the mean is
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iii. 99.7% of the data is within three standard deviations of the
mean, so the area representing the probability that a randomly
selected woman is within three standard deviation of the mean
.
is
Sidetrack: To represent 100% of the data, we would shade in the
entire curve, so what is the total area of any probability density
curve?
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(b) What percentage of women are 64 inches or taller?
What is the probability that a randomly selected woman will be
64 inches or taller? Be sure to write your result using probability
notation with an inequality.
(c) What is the probability that a randomly selected woman will be
between 61.3 and 66.7 inches tall? As always, be sure to show your
work by shading in the area corresponding to the probability in the
density curve and then be sure to write your result using probability
notation with an inequality.
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(d) What is the probability that a randomly selected woman will be
between 55.9 and 72.1 inches tall?
(e) What is the probability that a randomly selected woman will be
between 58.6 and 72.1 inches tall?
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(f) What is the probability that a randomly selected woman will be
shorter than 71 inches tall? Note, please stop after attempting this
problem.
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Here are the generic calculator steps for using your graphing calculator
to find the shaded area on a density curve that represents a probability
for a normally distributed random variable. How to use a TI-83 or TI-84
graphing calculator to find an area on a density curve that represents a
probability for a normally distributed random variable:
• Press 2nd and then Vars (to get the DISTR menu)
• Select normalcdf( (WARNING never select normalpdf)
• Complete the entry to obtain normalcdf(left bound, right bound,
Mean, Standard Deviation) by substituting in the appropriate values
and closing the parentheses. Press ENTER
Working together as a class, lets go back to the previous problem and
apply our new calculator skills to answer the question.
2. An average light bulb manufactured by the Acme Corporation lasts 300
days with a standard deviation of 50 days. Assuming that bulb life is
normally distributed.
(a) What is the probability that an Acme light bulb will last at most 365
days?
(b) What is the probability that an Acme light bulb will last at least
14 months (assume one month is 30 days)? Be sure to write the
probability notation with an inequality.
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(c) What is the probability that an Acme light bulb will last between 6
and 18 months (assume one month is 30 days)?
Now lets reverse the process. So far, weve been given data values and
used our calculators to find the probabilities. But now, well be given the
probabilities (i.e. the area of the shaded region on a normal density curve)
and use our TI-83 or TI-84 graphing calculators to find the data values
(i.e. the boundaries of our shaded regions). Here are the steps.
(a) Press 2nd and then Vars (to get the DISTR menu)
(b) Select invNorm(
(c) Complete the entry to obtain invNorm(Area of Shaded Region to the
Left, Mean, Standard Deviation) by substituting in the appropriate
values and closing the parentheses.
(d) Press ENTER
Note: invNorm will only give you the boundary (i.e. the data value) for
the shaded region to the left of the boundary. In other words invNorm
will find the boundary for the shaded region in either of the graph below.
−3
−2
|
0 X
−1
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1
2
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3. Suppose scores on an IQ test are normally distributed with a mean of 100
and a standard deviation of 10.
(a) Find the range of scores for the bottom 20% of students?
(b) The range of scores for the top 10% of students?
(c) The range of scores for the middle 70%?
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4. In a survey of men in the United States (ages 20 to 29), the mean height
was 69.9 inches with a standard deviation of 3.0 inches.
(a) What is the probability that a randomly selected man is shorter than
6 feet?
(b) How tall are the tallest 20% of men?
(c) What is the probability that a randomly selected man will be taller
than 5 feet 9 inches?
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(d) What height represents the 25th percentile?
(e) What is the probability that a randomly selected man is between 5
feet 9 inches and 6 feet 6 inches tall?
(f) How tall are the middle 75% of men?
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