Section 5.2
Summer 2013 - Math 1040
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Roadmap
Areas are probabilities!
Find probabilities for normally distributed random variables.
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Normal Curves
Ratios of the areas under different normal curves are equal when the
z-scores correspond. For instance, a z-score of 1 on the standard normal
curve corresponds to:
1
An x-value of 600 for a normal distribution with µ = 500 and
σ = 100.
2
1.75 years on a cell phone before switching for the normal distribution
with µ = 1.5 years and σ = 0.25 years.
3
A 57 minute trip to the supermarket for the normal distribution of
µ = 45 minutes and σ = 12 minutes.
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Steps to find Probabilities
1
Find the z-score for the x-value.
2
Use the standard normal table to find the area to the left of that
z-score.
3
If neccessary, find the area to the right, or repeat if given two
x-values.
The formula for the z-score is
z=
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x −µ
σ
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Example
# 11 Ford Fusion’s Breaking Distance.
Draw a normal curve when µ = 143 feet, σ = 5.12 feet. Find the
probability that a member is randomly selected and has breaking distance
between 145 feet and 155 feet.
The z-scores are:
145 − 143
≈ 0.39,
5.12
Their areas correspond to:
z1 =
0.6517,
z2 =
155 − 143
≈ 2.34.
5.12
0.9904
The difference is 0.9904 − 0.6517 = 0.3397. The probability that the
breaking distance is between 145 feet and 155 feet is 0.3397.
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Example
# 16. Adult male beagles have weights that are normally distributed with
a mean of 25 pounds and standard deviation of 3 pounds. A beagle is
selected at random. What is the probability that the weight is less than 23
pounds?
23 − 25
≈ −0.67
3
This corresponds to 0.2514. Then the probability for a randomly selected
beagle’s weight to be less than 23 pounds is 0.2514.
z=
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Example
Annual per capita consumption of coffee in the US is normally distributed.
For a random sample of size 30 from this population, a group of this size
consumes on average a mean of 24.2 gallons with standard deviation 1.479
gallons.
What is the probability that a randomly selected group of this size
consumes more than an average of 21.7 gallons?
21.7 − 24.2
≈ −1.69
1.479
This corresponds to 0.0455. We want the area to the right of this, so
1-0.0455 = 0.9505. The probability that a randomly selected group of 30
people will consume more than 21.7 gallons of coffee per year is 0.9505.
z=
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Assignments
Assignment:
1
Read pages 249 - 251.
Vocabulary: No new vocabulary.
Understand: How to find the probability of an event for any normal
random variable.
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