How Likely Are the Possible Values of a Statistic?

1
Section 6.4

• How Likely Are the Possible Values of a
  Statistic?
• The Sampling Distribution

2
Statistic

• Recall A statistic is a numerical summary of
  sample data, such as a sample proportion or a
  sample mean.

3
Parameter

• Recall A parameter is a numerical summary of a
  population, such as a population proportion or a
  population mean.

4
Statistics and Parameters

• In practice, we seldom know the values of
  parameters.
• Parameters are estimated using sample data.
• We use statistics to estimate parameters.

5
Example 2003 California Recall Election

• Prior to counting the votes, the proportion in
  favor of recalling Governor Gray Davis was an
  unknown parameter.
• An exit poll of 3160 voters reported that the
  sample proportion in favor of a recall was 0.54.

6
Example 2003 California Recall Election

• If a different random sample of about 3000 voters
  were selected, a different sample proportion
  would occur.

7
Example 2003 California Recall Election

• Imagine all the distinct samples of 3000 voters
  you could possibly get.
• Each such sample has a value for the sample
  proportion.

8
Statistics and Parameters

• How do we know that a sample statistic is a good
  estimate of a population parameter?
• To answer this, we need to look at a probability
  distribution called the sampling distribution.

9
Sampling Distribution

• The sampling distribution of a statistic is the
  probability distribution that specifies
  probabilities for the possible values the
  statistic can take.

10
The Sampling Distribution of the Sample Proportion

• Look at each possible sample.
• Find the sample proportion for each sample.
• Construct the frequency distribution of the
  sample proportion values.
• This frequency distribution is the sampling
  distribution of the sample proportion.

11
Example Sampling Distribution

• Which Brand of Pizza Do You Prefer?
• Two Choices A or D.
• Assume that half of the population prefers Brand
  A and half prefers Random D.
• Take a random sample of n 3 tasters.

12
Example Sampling Distribution

Sample No. Prefer Pizza A Proportion
(A,A,A) 3 1
(A,A,D) 2 2/3
(A,D,A) 2 2/3
(D,A,A) 2 2/3
(A,D,D) 1 1/3
(D,A,D) 1 1/3
(D,D,A) 1 1/3
(D,D,D) 0 0
13
Example Sampling Distribution

Sample Proportion Probability
0 1/8
1/3 3/8
2/3 3/8
1 1/8
14
Example Sampling Distribution
15
Mean and Standard Deviation of the Sampling
Distribution of a Proportion

• For a binomial random variable with n trials and
  probability p of success for each, the sampling
  distribution of the proportion of successes has
• To obtain these value, take the mean np and
  standard deviation for the
  binomial distribution of the number of successes
  and divide by n.

16
Example 2003 California Recall Election

• Sample Exit poll of 3160 voters.
• Suppose that exactly 50 of the population of all
  voters voted in favor of the recall.

17
Example 2003 California Recall Election

• Describe the mean and standard deviation of the
  sampling distribution of the number in the sample
  who voted in favor of the recall.
• µ np 3160(0.50) 1580

18
Example 2003 California Recall Election

• Describe the mean and standard deviation of the
  sampling distribution of the proportion in the
  sample who voted in favor of the recall.

BE VERY CAREFUL
19
The Standard Error

• To distinguish the standard deviation of a
  sampling distribution from the standard deviation
  of an ordinary probability distribution, we refer
  to it as a standard error.

20
Example 2003 California Recall Election

• If the population proportion supporting recall
  was 0.50, would it have been unlikely to observe
  the exit-poll sample proportion of 0.54?
• Based on your answer, would you be willing to
  predict that Davis would be recalled from office?

21
Example 2003 California Recall Election

• Fact The sampling distribution of the sample
  proportion has a bell-shape with a mean µ 0.50
  and a standard deviation s 0.0089.

22
Example 2003 California Recall Election

• Convert the sample proportion value of 0.54 to a
  z-score

23
Example 2003 California Recall Election
24
Example 2003 California Recall Election

• The sample proportion of 0.54 is more than four
  standard errors from the expected value of 0.50.
• The sample proportion of 0.54 voting for recall
  would be very unlikely if the population support
  were p 0.50.

25
Example 2003 California Recall Election

• A sample proportion of 0.54 would be even more
  unlikely if the population support were less than
  0.50.
• We there have strong evidence that the population
  support was larger than 0.50.
• The exit poll gives strong evidence that Governor
  Davis would be recalled.

26
Example 2003 California Recall Election

• Describe the mean and standard deviation of the
  sampling distribution of the proportion in the
  sample who voted in favor of the recall.

BE VERY CAREFUL
27
Summary of the Sampling Distribution of a
Proportion

• For a random sample of size n from a population
  with proportion p, the sampling distribution of
  the sample proportion has
• If n is sufficiently large such that the expected
  numbers of outcomes of the two types, np and
  n(1-p), are both at least 15, then this sampling
  distribution has a bell-shape.

28
Section 6.5

• How Close Are Sample Means to Population Means?

29
The Sampling Distribution of the Sample Mean

• The sample mean, x, is a random variable.
• The sample mean varies from sample to sample.
• By contrast, the population mean, µ, is a single
  fixed number.

30
Mean and Standard Error of the Sampling
Distribution of the Sample Mean

• For a random sample of size n from a population
  having mean µ and standard deviation s, the
  sampling distribution of the sample mean has
• Center described by the mean µ (the same as the
  mean of the population).
• Spread described by the standard error, which
  equals the population standard deviation divided
  by the square root of the sample size

31
Example How Much Do Mean Sales Vary From Week to
Week?

• Daily sales at a pizza restaurant vary from day
  to day.
• The sales figures fluctuate around a mean µ
  900 with a standard deviation s 300.

32
Example How Much Do Mean Sales Vary From Week to
Week?

• The mean sales for the seven days in a week are
  computed each week.
• The weekly means are plotted over time.
• These weekly means form a sampling distribution.

33
Example How Much Do Mean Sales Vary From Week to
Week?

• What are the center and spread of the sampling
  distribution?

34
Sampling Distribution vs. Population Distribution
35
Standard Error

• Knowing how to find a standard error gives us a
  mechanism for understanding how much variability
  to expect in sample statistics just by chance.

36
Standard Error

• The standard error of the sample mean
• As the sample size n increases, the denominator
  increase, so the standard error decreases.
• With larger samples, the sample mean is more
  likely to fall close to the population mean.

37
Central Limit Theorem

• Question How does the sampling distribution of
  the sample mean relate with respect to shape,
  center, and spread to the probability
  distribution from which the samples were taken?

38
Central Limit Theorem

• For random sampling with a large sample size n,
  the sampling distribution of the sample mean is
  approximately a normal distribution.
• This result applies no matter what the shape of
  the probability distribution from which the
  samples are taken.

39
Central Limit Theorem How Large a Sample?

• The sampling distribution of the sample mean
  takes more of a bell shape as the random sample
  size n increases. The more skewed the population
  distribution, the larger n must be before the
  shape of the sampling distribution is close to
  normal. In practice, the sampling distribution
  is usually close to normal when the sample size n
  is at least about 30.

40
A Normal Population Distribution and the Sampling
Distribution

• If the population distribution is approximately
  normal, then the sampling distribution is
  approximately normal for all sample sizes.

41
How Does the Central Limit Theorem Help Us Make
Inferences

• For large n, the sampling distribution is
  approximately normal even if the population
  distribution is not.
• This enables us to make inferences about
  population means regardless of the shape of the
  population distribution.