CHAPTER 6 Sampling Distributions

1

• CHAPTER 6 Sampling Distributions
• Homework 1abcd,3acd,9,15,19,25,29,33,43
• Sec 6.0 Introduction
• Parameter
• The "unknown" numerical values that is used to
  describe the properties of a population.
• Sample Statistics
• The computed numerical values from the
• measurements in a sample.

2

• Sec 6.1 What is a Sampling Distribution?
• Sampling Error
• The error results from using a sample instead
  of censusing the population to estimate a
  population quantity.
• Sampling Distributions
• The sample statistics vary from one sample to
  another. Therefore, there is a distribution
  function associated with each sampling statistic.
  This is called the sampling distribution.

3

• Example 6.1
• The three most popular numbers which were
  picked up by a group of students are 7, 17, and
  24. Assume that the population mean and standard
  deviation are 16 and 8, respectively.
• (a). List all the possible samples of two
  numbers that can be obtained from this three
  numbers.
• (b). Find the sample mean for each sample.
• (c). Compute the sample error for each sample.
• (Solutions in the note page)

4

• Example 6.2 (Basic)
• The population of average points per game (ppg)
  for the top five players of Orlando Magic is
  presented in the following table. (First eight
  games in 1996 playoff)
• Player ppg
• O'Neal(O) 25.3
• Anderson(A) 16.4
• Grant(G) 16.9
• Hardaway(H) 22.3
• Scott(S) 13.4
• (a) Find the sampling distribution of the mean of
  the
• "ppg" of three players from the population of
  five
• players. (Part (a) solution is in note page)

5

• Example 6.2 Continue
• (b) Compute the mean and standard deviation of
  the sample mean.
• ltSolution to part (b)gt
• Note If you forget how to do it, you need to
  review Chapter 4.

6

• Example 6.2 Continue
• (c) Find the sampling distribution of the sample
  median of the "ppg" of three players from the
  population of five players. (solution in note
  page)
• (d) Compute the mean and mean square error of the
  sample median.
• ltSolution to part (d)gt

7

• Sec 6.2 Some Criterions for choosing a
• Statistics
• Point Estimator
• A point estimator of a population parameter is
  a rule that tell us how to obtain a single number
  based on the sample data. The resulting number
  is called a point estimator of this unknown
  population parameter.

8

• Unbiasedness
• The mean of the sampling distribution of a
• statistic is called the expectation of this
  statistic. If the expectation of a statistics is
  equal to the population parameter this statistics
  is intended to estimate, then the statistics is
  called an unbiased estimator of this population
  parameter. If the expectation of a statistics is
  not equal to the population parameter, the
  statistics is a biased estimator.

9

• NOTES
• (1). If the population is normally distributed
  and the sample are randomly selected from this
  population, the sample mean is the best unbiased
  estimator of the population mean.
• (2). If the population has an extremely skewed
  distribution and the sample size of the random
  sample is small, the sample mean may not be best
  estimator of the population mean.

10

• Example 6.3
• Consider the probability distribution shown
  here.
• x p(x)
• 2 1/3
• 4 1/3
• 9 1/3
• (a) Find m.
• ltSolution to (a)gt

11

• Example 6.3 Continue
• (b) List all possible samples of three
  observations.
• ltsolution for part (b)gt
• (2,2,2)
• (2,,2,4) (2,4,2) (4,2,2)
• (2,4,4) (4,2,4) (4,4,2)
• (4,4,4)
• (2,2,9) (2,9,2) (9,2,2)
• (2,4,9) (2,9,4) (4,2,9) (4,9,2) (9,2,4) (9,4,2)
• (4,4,9) (4,9,4) (9,4,4)
• (2,9,9) (9,2,9) (9,9,2)
• (4,9,9) (9,4,9) (9,9,4)
• (9,9,9)

12

• Example 6.3 Continue
• (c) For a sample of 3 observations, find the
  sampling distribution of the sample mean and its
  mean value
• ltSolution to part (c)gt
• mean P(mean)
• 2 1/27
• 8/3 3/27
• 10/3 3/27
• 4 1/27
• 13/3 3/27
• 5 6/27
• 17/3 3/27
• 20/3 3/27
• 22/3 3/27
• 9 1/27

13

• Example 6.3 Continue
• (d) For a sample of 3 observations, find the
  sampling distribution of the sample median and
  E(median).
• ltSolution to part (d)gt
• median P(median)
• 2 7/27
• 4 13/27
• 9 7/27
• E(median) Smedian P(median)
• 2 7/27 413/27 97/27 14.33

14

• Sec 6.3 Central Limit Theorem
• Some properties of the sample mean
• (1). The expectation of the sampling distribution
  of is equal to
  the population mean m, i.e.
• (2). The sampling distribution of sample mean is
  approximately normal for sufficiently large
  sample size even if the sampled population is not
  normally distributed.
• (3). The sampling distribution of sample mean is
  exactly normal if the sampled population is
  normally distributed no matter how small the
  sample size is.

15

• Central Limiting Theorem
• Suppose a random sample of size n is drawn from
  a population. If the sample size n is
  sufficiently large, then the sample mean has at
  least approximately a normal distribution.
• The mean value of the sample mean is always
  equal to the population mean, and the standard
  error of the sample mean is always equal to
• where s is the population standard deviation.

16

• Note Each of the following figure shows the
  population distribution (n1) and three sampling
  distributions of the sample mean for different
  populations. We can see that the sampling
  distribution of the sample mean is approximately
  normal distributed when the sample size getting
  larger.

17
Figure 6.1 Normal Population
2.0
2.0
f(x)
f(x)
1.0
1.0
0.0
0.0
-4
-2
0
2
4
-4
-2
0
2
4
n 1
n 2
2.0
2.0
f(x)
f(x)
1.0
1.0
0.0
0.0
-4
-2
0
2
4
-4
-2
0
2
4
n 4
n 30
18
Figure 6.2 Chi-Square Population
2.0
2.0
f(x)
f(x)
1.0
1.0
0.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
n 1
n 2
2.0
2.0
f(x)
f(x)
1.0
1.0
0.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
n 4
n 30
19
Figure 6.1 Uniform Population
8
8
6
6
f(x)
4
f(x)
4
2
2
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
n 1
n 2
8
8
6
6
f(x)
4
f(x)
4
2
2
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
n 4
n 30
20

• Example 6.4
• Suppose that you selected a random sample of
  size 4 from a normal population with mean 8 and
  standard deviation 2.
• (a) Is the sample mean normally distributed?
  
• Explain.
• (b) Find P( gt 10).
• (c) Find P( lt 7).

21

• ltSolutions to EX 6.4gt
• (a) Yes. The sampling distribution of sample mean
  is exactly normal if the sampled population is
  normally distributed no matter how small the
  sample size is.
• standard error
• (b) P(x gt 10)
• P(z gt 2) 0.0228
• (c) P(x lt 7)
• P(z lt -1) 0.1587.

22

• Example 6.5
• As reported in the National Center for Health
  Statistics, males who are six feet tall and
  between 18 and 24 years old have a mean weight of
  175 pounds with a 14 pound standard deviation.
• (a). Find the probability that a random sample
  of 196 males who are six feet tall and between 18
  and 24 years of age has a mean weight greater
  than 176 pounds.
• (b). Find the probability that a random sample
  of 4 males who are six feet tall and between 18
  and 24 years of age has a mean weight greater
  than 168 pounds. State the necessary
  assumptions.

23

• ltSolutions to EX 6.5gt
• (a) standard error
• P( x gt 176)
• P( z gt 1) 0.1587
• (b) standard error
• P( x gt 168)
• P( z gt -1) 0.8413.
• The population needs to have normal distribution.