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Math 011 – CHAPTER 6 Normal Probability Distributions
6-2 Standard Normal Distribution
DEFINITION
A continuous random variable has a uniform distribution if
Note:
Example 1: For New York City weekday late-afternoon subway travel from Times Square to the
Mets Stadium, you can take the #7 train that leaves Times Square every 5 minutes. Given the
subway departure schedule and the arrival of a passenger, the waiting time x is between 0 min
and 5 min. All the different possible waiting times are equally likely.
a. Graph the uniform distribution.
Density Curve
Density Curve Requirements
Example 1: (continued)
b. Given the uniform distribution in part (a), find the probability that a randomly selected
passenger has a waiting time of less than 2 minutes.
c. Find the probability that a randomly selected passenger has a waiting time greater than
3 minutes.
The standard normal distribution
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Finding Probabilities when Z-scores are given
Example 2: A bone mineral density test can be helpful in identifying the presence or likelihood
of osteoporosis, a disease causing bone to become more fragile and more likely to break. The
result of a bone density test is commonly measured as z scores. The population of z scores is
normally distributed with a mean of 0 and standard deviation of 1. Test results meet the
requirements of a normal standard distribution. A randomly selected adult undergoes a bone
density test.
a. Find the probability that test result is a reading less than 1.55.
b. Find the probability that the test result is above -1.00 (A value above -1.00 is considered
to be in the “normal” range of bone density readings.)
c. A bone density test reading between -1.00 and -2.50 indicates that the subjects has
osteopenia, which is some bone loss. Find the probability that a randomly selected
subject has a reading between -1.00 and -2.50.
Notation
Finding z scores from known areas:
Example: (Bone Density Example continued)
d. Find the bone density score corresponding to the P95, the 95th percentile.
e. Find the bone density test score that separates the bottom 3% and find the bone
density test score that separates the top 3%.
Critical value
Notation
Example: In the expression zα, let α 0.03. Find zα.
6-3 Applications of Normal Distribution
Procedure for finding areas with a nonstandard normal distribution
Example 1: The social organization Tall Clubs International has a requirement that women must
be at least 70 in tall. Given that women have normally distributed heights with a mean of 63.8in
and a standard deviation of 2.6in, find the percentage of women who satisfy that height
requirement.
Example 2: British Airways and many other airlines have a requirement that a member of the
cabin crew must have a height between 62in and 73in. Given that women have normally
distributed heights with a mean of 63.8in and a standard deviation of 2.6in, find the percentage
of women who satisfy that height requirement.
Example 3: When designing an environment, one common criterion is to use a design that
accommodates 95% of the population. What aircraft ceiling height will allow 95% of women to
stand without bumping their heads? That is, find the 95th percentile of the heights of women.
Assume that heights of women are normally distributed with a mean of 63.8in and standard
deviation of 2.4in.
Example 4: Assume that students are to be separated into a group with IQ scores in the bottom
30%, a second group with IQ scores in the middle 40%, and a third group with IQ scores in the
top 30%. The Wechsler Adult Intelligence Scale yields an IQ score obtained through a test, and
the scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the
Wechsler IQ scores that separate the three groups.
Example 5: Find the area of the shaded region. The graph depicts IQ scores of adults, and those
scores are normally distributed with a mean of 100 and a standard deviation of 15.
6-4 Sampling Distributions and Estimators
Sampling Distribution of a statistic
Sampling distribution of the sample mean
Properties
Sampling distribution of the variance
Properties
Sampling distribution of the proportion
Properties
Unbiased estimators
Biased estimators
6-5 The Central Limit Theorem
The Central Limit Theorem
Given:
Conclusions:
Practical Rules Commonly Used
Notation:
Example:
Suppose an elevator has a maximum capacity of 16 passengers with a total weight of 2500 lb.
Assuming a worst case scenario in which the passengers are all male, what are the chances the
elevator is overloaded?
Assume male weights follow a normal distribution with a mean of 182.9 lb and a standard
deviation of 40.8 lb.
a.
Find the probability that 1 randomly selected male has a weight greater than 156.25 lb.
b.
Find the probability that a sample of 16 males have a mean weight
greater than
156.25 lb (which puts the total weight at 2500 lb, exceeding the maximum capacity).