Mean and s.d. of sample means

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Lesson 13b
Distribution of Sample Means

• Mean and s.d. of sample means
• Central limit theorem
• Margin of error

Main Ideas
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• 1. Sample Mean is a Variable
• If different people were to take random samples
  they would probably get different results.

Suzies sample
Sample mean is 3.2
Population mean is m 3.6
Sams sample
Sample results vary
Sample mean is 3.8
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• 2. Distribution of the Sample Means
• Population has mean m and s.d. s, sample of size
  n is selected randomly from population.
• For large n, sample means have an approximate
  normal distribution.
• That is, sample means vary in a bell-shaped
  fashion about the population mean.

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• 3. Mean and S.D. of Sample Means
• Mean of sample means m (population mean)
• s.d. of sample means
• (pop. s.d. divided by square root of sample
  size).

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• 4. Central Limit Theorem
• The central limit theorem says that sample means
  have an approximate normal distribution for large
  sample sizes regardless of the population
  distribution.

Approximate. normal dist. of sample means
Mean of sample means m, s.d. of sample mean
s/on
Pop. mean m, pop. s.d. s
Pop. dist. has arbitrary shape
m
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Example 1. A machine that puts soft drink into
bottles varies in the amount it dispenses because
of random factors. If the machine is working
o.k., the mean should be 20 ounces and the s.d.
is 0.4 ounces. A quality control inspector
takes a sample of 25 soft drink bottles and
finds an sample mean of 19.8 ounces. Is a
mean of 19.8 unusually small, or is this to be
expected given the random variability of the
process? We will answer by computing the
probability that a sample mean with n 25 is
19.8 or less.
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Example 1. (continued). If process is o.k,
then mean of sample means 20 s.d. of sample
means 0.4/sqrt(25) .08 Using the normal
probability tables, we find P( sample mean lt
19.8) P Z lt (19.8 - 20)/.08 P(Z lt
-2.50) .0062 Thus 19.8 is an unusually small
sample mean for a sample of 25. This is an
indication that the machine is defective and not
putting enough liquid into the bottles.
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• 5. How Close is the Sample Mean to the
  Population Mean?
• The sample mean has approximately a 95 chance
  of falling in the interval

Margin of error
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• Example 2. Suppose the weights of college-age
  males have a mean of 155 lbs. and a s.d. of 10
  lbs. We will compare margins of error of the
  sample mean for samples of n 30 and n 120.
• For n 30, the margin of error is
• 1.9610/sqrt(30) 3.6
• The sample mean is likely to differ from the
  pop. mean by no more than 3.6 inches when n 30.
• For n 120, the margin of error is
• 1.9610/sqrt(120 1.8
• We have to take 4 times the sample to cut the
  margin of error in half.

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• 6. Factors Affecting the Margin of Error of
  Sample Mean
• The margin of error of the sample mean is
• The margin of error will be smaller for
  populations with smaller standard deviations.
• Margin of error can be made as small as we like
  by increasing the sample size.

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We now know that the sample means have an approx.
normal distribution regardless of the population
distribution. What happens when the population
itself has a normal distribution?
In this case, there is no approximate about it.
The distribution of the sample means will have a
normal distribution for any sample size, not just
large samples.
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