Transcript Chapter 9 Day 4

CHAPTER 9 DAY 4
Warm - Up
 Harley Davidson motorcycles make up 14% of all
motorcycles registered in the U.S. You plan to
interview an SRS of 500 motorcycle owners.
 What is the approximate distribution of your sample
that own Harleys? Standard deviation?
 Why can you “do this”? (use Rules of Thumb)
 How likely is your sample to contain 20% or more
who own Harleys?
 How likely is your sample to contain at least 15%
who own Harleys?
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 μ and its
standard deviation is σ/√n
 The behavior of x in repeated samples is much
like that of the sample proportion
 The sample mean
x is an unbiased estimator of the
population mean μ.
 The values of x are less spread out for larger
samples. Their standard deviation decreases at the
rate √n, so you must take a samples 4 times as large
to cut the standard deviation of x in half.
 You should only use the recipe for standard deviation
when the population is at least 10 times as large as
the sample.
Example
 The height of young women varies
approximately according to the N(64.5,2.5)
distribution. If we choose an SRS of 10 young
women, find the mean and standard
deviation of the sample.
Central Limit Theorem
 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
 In other words, as sample size increases the
distribution becomes more normal.
Example
 A company that owns a fleet of cars for its
sales force has found that the service lifetime
of disc brake pads varies form car to car
according to a normal distribution with mean
μ= 55,000 and standard deviation σ = 4500
miles. The company installs a new brand of
brake pads on 8 cars.
 If the new brand has the same lifetime
distribution as the previous type, what is the
distribution of the sample mean lifetime for the
8 cars?
 The average life of the pads on these 8 cars turns
out to be x = 51,800 miles. What is the
probability that the sample mean lifetime is
51,800 miles or less if the lifetime distribution is
unchanged? The company takes this probability
as evidence that the average lifetime of the new
brand of pads is less than 55,000 miles.
Remember the Law of Large
Numbers?
 Draw observations at random from any
population with finite mean μ. As the number
of observations drawn increases, the mean x
of the observed values gets closer and closer
to μ.