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AP Statistics
9.3 Sample Means
Date _________________________
When we record quantitative variables such as household income, heights of adult females, or a patient’s heart
rate we are interested in statistics such as the mean, median, or standard deviation of the variable. Sample
means are, perhaps, the most common of these statistics and we will turn our attention to the sampling
distribution of the mean of the responses in a SRS.
Mean and Standard Deviation of a Sample Mean
Suppose that x is the mean of an SRS of size n drawn from a large population with mean  and standard
deviation  . Then
 The mean of the sampling distribution of x is
 And the standard deviation of x is
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A couple points to keep in mind
about
the
behavior of x :
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1.
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2.
3.
4.
Example 1 ACT Scores: The scores of individual students on the American College Testing composite college
entrance examination have a normal distribution with mean 18.6 and standard deviation of 5.9.
a) What is the probability that a single student randomly chosen from all those taking the test scores 21 or
higher?
b) Now take an SRS of 50 students who took the test. What are the mean and standard deviation of the
average (sample mean) score for the 50 students. Do your results depend on the fact that individual
scores are normally distributed?
c) What is the probability that the mean score

x
of these students is 21 or higher?
Example 2 Men’s’ Heights: The heights of American males are N (69, 2.5).
a) What is the probability that a single man will be six feet tall or taller?
b) Now consider an SRS of 10 men. What is the probability that the mean height of these men
tall or taller?
x
is six feet
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Central Limit Theorem:
Example 3 Traffic Accidents: The number of traffic accidents per week at an intersection varies with mean 2.2
and standard deviation 1.4. The number of accidents in a week must be a whole number so the population
distribution is not normal.
a) Let x be the mean number of accidents per week at the intersection during a year (52 weeks). What is
the approximate distribution of x according the Central Limit Theorem?
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b) What is the approximate probability that x is less than 2?
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c) What is the approximate probability that there are fewer than 100 accidents at the intersection in a year?
Example 4 More Auto Accidents: A study of rush-hour traffic in San Francisco counts the number of people in
each car entering a freeway at a suburban interchange. Suppose that the count has a mean of 1.5 and standard
deviation of 0.75.
a) Traffic engineers estimate that the capacity of the interchange is 700 cars per hour. According to the
Central Limit Theorem, what is the approximate distribution of the mean number of persons x in 700
randomly selected cars at this interchange?
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b) What is the probability that 700 cars will carry more than 1075 people?