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Normal Probability
Distributions
Intro to Normal
Distributions & the
STANDARD Normal
Distribution
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1. mean, median, and mode are equal
2. bell shaped and symmetric about the
mean
3. total area under the curve is 1
4. the curve approaches, but never
touches the x axis as it extends away
from the mean
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A Normal distribution with
 mean
= 0 and
 standard deviation = 1
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The cumulative area is close to 0 for
z-scores close to z = -3.49
The cumulative area increases as the
z-score increases.
The cumulative area for z = 0 is 0.5000
The cumulative area is close to 1 for
z-scores close to z = 3.49
1.
2.
3.
4.
Sketch the curve and shade the
appropriate area under the curve.
To find the area LEFT of z, find the area
that corresponds to z on the table.
To find the area RIGHT of z, find the
area that corresponds to z on the table,
then subtract from 1.
To find the are BETWEEN 2 z-scores,
find each area, then subtract.
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1.
2.
3.
4.
5.
left of z = -1.02
right of z = 2.30
left of z = 0.45
right of z = - 1.99
between z = -2.34 and z = 1.89
Normal Distributions:
Finding Probabilities
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1. Sketch a normal curve.
2. Find the z-score Recall: z = x - µ
σ
3. Plot your z-score and shade as indicated.
4. Find the area under the curve using the
standard normal distribution table (table #4)
5. Interpret the results to answer the
question.
Find P(x < 200)
Find P(x > 155)
Find P(172 < x < 192)
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13. A survey was conducted to measure the
heights of US men. In the survey,
respondents were grouped by age. In the 2029 age group, the heights were normally
distributed, with a mean of 69.9 inches and a
standard deviation of 3.0 inches. A study
participant is randomly selected. Find each
prob:
A) his height is less than 66 in.
B) his height is between 66 and 72 in.
C) his height is more than 72 in.
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20. The times per workout an athlete uses a
stairclimber are normally distributed, with a
mean of 20 minutes and a standard deviation
of 5 minutes. An athlete is randomly selected.
Find each probability.
A) the athlete uses a stairclimber for less than
17 minutes.
B) the athlete uses a stairclimber between 20
and 28 minutes.
C) the athlete uses a stairclimber for more
than 30 minutes.
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26. Monthly utility bills are
normally distributed with a mean
of $100 and a standard deviation of
$12.
A) What percent of the utility bills
are more than $125?
B) If 300 utility bills are randomly
selected, about how many would
you expect to be less than $90?