Sampling Distribution of a Statistic

1
Chapter 8

• Sampling Distribution of a Statistic

2
Binomial Distribution

• A binomial random variable X is the total number
  of successes in n independent Bernoulli trials,
  on which each trial, the probability of success
  is p. We say X is B(n,p).

3
Mean and Standard Deviation of a Binomial
Distribution
4
Approximating the Binomial with the Normal

• We can use the normal distribution to approximate
  the binomial when np 5 and nq 5.
• If X is B(n, p) and np 5 and nq 5 then X can
  be approximated by

5
Homework 15

• Read pages 499-504, 509-510, 522, 525
• LDI 8.5, 8.6
• EX 8.1, 8.4, 8.7, 8.8, 8.11, 8.12

6
Definition

• The sampling distribution of a statistic is the
  distribution of values of the statistic in all
  possible samples of the same size n taken from
  the same population.

7
Proportion of Women

• In rural China, many villages are experiencing a
  lack of women. This suggests the the proportion
  of women in the population is less than 50.
• We want to estimate the proportion of women in
  rural villages in China, so well take a sample

8
Sample Proportion (statistic)
9
Population Proportion (parameter)
10
Hit List
11
What Do We Expect of Sample Proportions?

• The values of the sample proportion vary from
  random sample to random sample in a predictable
  way.
• The shape of the distribution of the sample
  proportion is approximately symmetric and
  bell-shaped.

12
What Do We Expect of Sample Proportions?

• The center of the distribution of the sample
  proportion values is at the true population
  proportion p.
• With a larger sample size n, the sample
  proportion values tend to be closer to the true
  population proportion p. The values vary less
  around p.

13
The definition of p-hat

• Let x be the number of successes out of n trials
  then
• Also recall the definition of m and s for a
  binomial distribution

14
Do the Linear Transformation
And
15
Mean and Standard Deviation of a Binomial
Distribution
16
Mean and Standard Deviation of p-hat
17
Normal Approximation of Binomial

• We can use the normal distribution to approximate
  the binomial when np 5 and nq 5.
• If X is B(n, p) and np 5 and nq 5 then X can
  be approximated by

18
Normal Approximation of p-hat

• If n is sufficiently large (np5 and nq5), the
  distribution of p-hat will be approximately
  normal.

19
Example 8.3

• Suppose of all voters in a state, 30 are in
  favor of Proposition A.
• If we sample 400 voters what is the probability
  that less than 25 will be in favor of
  proposition A?
• What is the probability that the proportion of
  voters will be between 27 and 33?

20
68-95-99.7 Rule

• What percentage of p-hats fall within 2 standard
  deviations of the mean p?
• About 95 of all random samples should result in
  a sample proportion p-hat that is within two
  standard deviations of the population proportion
  p.

21
Works Both Ways

• If 95 of the p-hats are within 2 standard
  deviations of p then 95 of the time p should be
  within an interval that is 2 standard deviations
  from p-hat.
• Standard Error

22
Standard Error

• In practice we do not have the population
  standard deviation of the sampling statistic. So,
  we have to estimate it with the standard error.
  In this case it is an estimate of the average
  distance of possible p-hat values from the
  population proportion p.

23
Basic Idea

• We are quite confident that the true population
  proportion is in the interval that is plus or
  minus two standard errors of p-hat

24
What it does not say!

• Note again that the 95 here is a probability
  associated with the method. We say that 95 of
  the time the interval will work in capturing the
  true value of p. But once we have an interval,
  there is no more discussion about the probability
  of the parameter being contained in the interval.

25
Lets Do It

• LDI 8.5
• LDI 8.6

26
Homework 16

• Read pages 531-532, 534-541, 543-544
• LDI 8.8, 8.9, 8.10, 8.11
• EX 8.17, 8.18, 8.22, 8.25, 8.28, 8.30

27
Sampling Distribution of the Mean (x-bar)

• The sampling distribution of the mean is the
  distribution of values of the sample mean in all
  possible random samples of the same size n taken
  from the same population.

28
Sampling Distribution of the Mean (x-bar)

• The distribution of x-bar will be approximately
  normal if the sample size is large enough no
  matter what the original distributions shape.
• If the original distribution is normal, then the
  distribution of x-bar will be exactly normal

29
Lets Do It!

• Lets simulate the distribution of x-bar using
  these programs. Well use XBARINT for LDI 8.8.
• Well then try it again using the AGE data as our
  distribution to sample from. That is done using
  XBARSIM
• LDI 8.9

30
Sampling Distribution of the Mean

• If the original distribution has mean m and
  standard deviation s, then for large enough
  samples the distribution of x-bar will be
  approximately

31
Sampling Distribution of the Mean

• If the original distribution is normal with mean
  m and standard deviation s, then the distribution
  of x-bar will be exactly

32
The Standard Error of the Mean (SEM)

• Again, we will rarely know the population
  standard deviation of the mean ( ) we
  instead have to estimate it using the sample
  standard deviation s. We replace ? with s to get
  an estimate standard deviation of x-bar.

33
Lets Do It

• LDI 8.10
• LDI 8.11