AP STATISTICS LESSON 9 - 3

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					AP STATISTICS
LESSON 9 - 3

  SAMPLE MEANS
ESSENTIAL QUESTION:
How are questions involving
sample means solved?
 Objectives:

  To find the mean of a sample when μ is
   unknown.
  To find the standard deviation when the σ is
   unknown.
  To solve problems involving sample means.
          Sample Means
Sample proportions arise most often when
we are interested in categorical variables.

We then ask questions like “What proportion
of U.S. adults have watched Survivor II?”

Because sample means are just averages of
observations, they are among the most
common statistic.
    Example 9.9       page 514
    Bull Market or Bear Market?
Page 515 histograms show:
   Averages are less variable
    than individual observations.

 More detailed examination
 of the distribution would
 point to a second principle:
   Averages are more normal
    than individual observations.
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 μ and
    standard deviation σ. Then the
    mean of the sampling distribution of
    x is μx = μ and its standard deviation
    is σ = σ/ √ n.
Mean and Standard Deviation of a
     Sample Mean (continued…)
 The sample mean x is an unbiased
  estimate of the population mean μ.
 The values of x are less spread out for
  larger samples.
 You should only use the recipe
 σ /√ n for the standard deviation of x
  when the population is at 10 times as
  large as the sample.
     Example 9.10 Page 516
      Young Women’s Heights
The height of a young women varies
approximately according to N(64.5, 2.5 )
distribution.

What is the probability that a randomly
selected woman is taller than 66.5 inches?

What is the probability of an SRS of 10 young
women being greater than 66.5?
Sampling Distributions of a Sample
 Mean from a Normal Population

 Draw an SRS of size n from a population
 that has the normal distribution with
 mean μ and standard deviation σ.

 Then the sample mean x has the
 normal distribution N(μ,σ/√n ) with
 mean μ and standard deviation σ/√ n .
   The Central Limit Theorem

The central limit theorem states that when an
infinite number of successive random
samples are taken from a population, the
distribution of sample means calculated for each
sample will become approximately normally
distributed with mean μ and standard deviation
σ / √ n (N(μ, σ / √ n)) as the sample size (N)
becomes larger, irrespective of the shape of
the population distribution.
Central Limit Theorem           (continued…)
Draw an SRS of size n from any population
 whatsoever with mean μ and finite standard
 deviation σ. When n is large, the sampling
 distribution of the sample mean x is close to
 the normal distribution N(μ,σ/√ n ) with mean
 μ and standard deviation σ√ n

 How large a sample size n is needed for x to
 be close to normal depends on the
 population distribution. More observations
 are required if the shape of the population
 distribution is far from normal.
          Example 9.12 page 521
           Exponential Distribution
The distribution is strongly right-
skewed, and the most probable
outcomes are near 0 at one end
of the range of possible values.

 The mean μ of this distribution is
1 and its standard deviation σ is
also 1. This particular distribution
is called an exponential
distribution from the shape of its
density curve.

As n increases, the shape
becomes more normal.
       Example 9.13 Page 522
       Servicing Air Conditioners
 The time that a technician
requires to perform
preventive maintenance on
an air conditioning unit is
governed by the
exponential distribution
whose density. The mean
time is μ = 1 hour and the
standard deviation is σ = 1
hour. Your company
operates 70 of these units.

What is the probability that
their average maintenance
time exceeds 50 minutes?
      Sampling Distribution
The Sampling distribution of a sample mean x
has a mean μ and standard deviation σ√ n .
The distribution is normal if the population
distribution is normal; it is approximately
normal for large samples in any case.