Distribution%20of%20the%20Sample%20Mean

1
Lesson 8 - 1

• Distribution of the Sample Mean

2
Objectives

• Understand the concept of a sampling distribution
• Describe the distribution of the sample mean for
  sample obtained from normal populations
• Describe the distribution of the sample mean from
  samples obtained from a population that is not
  normal

3
Vocabulary

• Statistical inference using information from a
  sample to draw conclusions about a population
• Standard error of the mean standard deviation
  of the sampling distribution of x-bar

4
Conclusions regarding the sampling distribution
of X-bar

• Shape normally distributed
• Center mean equal to the mean of the population
• Spread standard deviation less than the
  standard deviation of the population
• Law of Large numbers tells us that as n increases
  the difference between the sample mean, x-bar and
  the population mean µ approaches zero

5
Central Limit Theorem
X or x-barDistribution
Regardless of the shape of the population, the
sampling distribution of x-bar becomes
approximately normal as the sample size n
increases. Caution only applies to shape and
not to the mean or standard deviation
Random Samples Drawn from Population
Population Distribution
6
Summary of Distribution of x
Shape, Center and Spread of Population Distribution of the Sample Mean Distribution of the Sample Mean Distribution of the Sample Mean
Shape, Center and Spread of Population Shape Center Spread
Normal with mean, µ and standard deviation, s Regardless of sample size, n, distribution of x-bar is normal µx-bar µ s sx-bar ------- ?n
Population is not normal with mean, µ and standard deviation, s As sample size, n, increases, the distribution of x-bar becomes approximately normal µx-bar µ s sx-bar ------- ?n
7
What Happens to Sample Spread

• If the random variable X has a normal
  distribution with a mean of 20 and a standard
  deviation of 12
• If we choose samples of size n 4, then the
  sample mean will have a normal distribution with
  a mean of 20 and a standard deviation of 6
• If we choose samples of size n 9, then the
  sample mean will have a normal distribution with
  a mean of 20 and a standard deviation of 4

8
Example 1

• The height of all 3-year-old females is
  approximately normally distributed with µ 38.72
  inches and s 3.17 inches. Compute the
  probability that a simple random sample of size n
  10 results in a sample mean greater than 40
  inches.

P(x-bar gt 40)
µ 38.72 s 3.17 n 10
sx 3.17 / ?10 1.00244
x - µ Z ------------- sx
1.28 ----------------- 1.00244
40 38.72 ----------------- 1.00244
1.277
normalcdf(1.277,E99) 0.1008 normalcdf(40,E99,38
.72,1.002) 0.1007
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Example 2

• Weve been told that the average weight of
  giraffes is 2400 pounds with a standard deviation
  of 300 pounds. Weve measured 50 giraffes and
  found that the sample mean was 2600 pounds. Is
  our data consistent with what weve been told?

P(x-bar gt 2600)
µ 2400 s 300 n 50
sx 300 / ?50 42.4264
x - µ Z ------------- sx
200 ----------------- 42.4264
2600 2400 ----------------- 42.4264
4.714
normalcdf(4.714,E99) 0.000015 normalcdf(2600,E9
9,2400,42.4264) 0.0000001
10
Summary and Homework

• Summary
• The sample mean is a random variable with a
  distribution called the sampling distribution
• If the sample size n is sufficiently large (30 or
  more is a good rule of thumb), then this
  distribution is approximately normal
• The mean of the sampling distribution is equal to
  the mean of the population
• The standard deviation of the sampling
  distribution is equal to s / ?n
• Homework
• pg 431 433 3, 4, 6, 7, 12, 13, 22, 29

11
Homework

• 3) standard error of the mean
• 4) zero
• 6) population is normal
• 7) four
• 12) µ64, s 18, n36 µx-bar 64, sx-bar
  18/?36 18/6 3
• 13) µ64, s 18, n36 µx-bar 64, sx-bar
  18/?36 18/6 3
• 22) µ81.7, s 6.9
• a) P(x lt 75) 0.1658
  normalcdf(-E99,75,81.7,6.9)
• b) n5 P(xlt75) 0.01496
  normalcdf(-E99,75,81.7,6.9/?5)
• c) n8 P(xlt75) 0.00301
  normalcdf(-E99,75,81.7,6.9/?8)
• d) very unlikely (or unusual)
• 29) P(x lt 45) 0.0078
  normalcdf(-E99,45,50,16/?60)