Topic VI: Sampling Distributions

1
Topic VI Sampling Distributions
2
Importance of Samples and Sampling

• Sampling is important in Statistics because
  statistics which are calculated from samples are
  used to make estimates about the values of
  population parameters.

3
Sampling Distributions
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
4
Sampling Distributions

• A sampling distribution is a distribution of all
  of the possible values of a statistic for a given
  sample size selected from a population

5
Developing a Sampling Distribution

• Assume there is a population
• Population size N4
• Random variable, X,is age of individuals
• Values of X 18, 20,22, 24 (years)

D
C
A
B
6
Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution
P(x)
.3
.2
.1
0
x
18 20 22 24 A
B C D
Uniform Distribution
7
Now consider all possible samples of size n2
Developing a Sampling Distribution
(continued)
16 Sample Means
16 possible samples (sampling with replacement)
8
Sampling Distribution of All Sample Means
Developing a Sampling Distribution
(continued)
Sample Means Distribution
16 Sample Means
_
P(X)
.3
.2
.1
_
0
18 19 20 21 22 23 24
X
(no longer uniform)
9
Summary Measures of this Sampling Distribution
Developing a Sampling Distribution
(continued)
10
Comparing the Population with its Sampling
Distribution
Population N 4
Sample Means Distribution n 2
_
P(X)
P(X)
.3
.3
.2
.2
.1
.1
_
0
0
X
18 19 20 21 22 23 24
18 20 22 24 A
B C D
X
11
Sampling Distribution of the Mean
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
12
Standard Error of the Mean

• Different samples of the same size from the same
  population will yield different sample means
• A measure of the variability in the mean from
  sample to sample is given by the Standard Error
  of the Mean
• (This assumes that sampling is with replacement
  or
• sampling is without replacement from an infinite
  population)
• Note that the standard error of the mean
  decreases as the sample size increases

13
If the Population is Normal

• If a population is normal with mean µ and
  standard deviation s, the sampling distribution
  of is also normally distributed with
• and

14
Z-value for Sampling Distributionof the Mean

• Z-value for the sampling distribution of

where sample mean population mean
population standard deviation n
sample size
15
Sampling Distribution Properties
Normal Population Distribution

• (i.e. is unbiased )

Normal Sampling Distribution (has the same mean)
16
Sampling Distribution Properties
(continued)

• As n increases,
• decreases

Larger sample size
Smaller sample size
17
If the Population is not Normal

• We can apply the Central Limit Theorem
• Even if the population is not normal,
• sample means from the population will be
  approximately normal as long as the sample size
  is large enough.
• Properties of the sampling distribution
• and

18
Central Limit Theorem
the sampling distribution becomes almost normal
regardless of shape of population
As the sample size gets large enough
n?
19
Effect of Sample Size on Sampling Distribution
20
Sampling Distributions I

• The Central Limit Theorem
• If x1, x2, x3, , xn is a random sample taken
  from a population with mean µ and variance s2,
  then
• E( ) µ
• Var( ) s2/n
• Where is the sample mean

21
If the Population is not Normal
(continued)
Sampling distribution properties
Population Distribution
Central Tendency
Sampling Distribution (becomes normal as n
increases)
Variation
Larger sample size
Smaller sample size
22
How Large is Large Enough?

• For most distributions, n gt 30 will give a
  sampling distribution that is nearly normal
• For fairly symmetric distributions, n gt 15
• For normal population distributions, the sampling
  distribution of the mean is always normally
  distributed

23
Example

• Suppose a population has mean µ 8 and standard
  deviation s 3. Suppose a random sample of size
  n 36 is selected.
• What is the probability that the sample mean is
  between 7.8 and 8.2?

24
Example
(continued)

• Solution
• Even if the population is not normally
  distributed, the central limit theorem can be
  used (n gt 30)
• so the sampling distribution of is
  approximately normal
• with mean 8
• and standard deviation

25
Example
(continued)

• Solution (continued)

Sampling Distribution
Standard Normal Distribution
Population Distribution
.1554 .1554
?
?
?
?
?
?
?
?
?
?
Sample
Standardize
?
?
-0.4 0.4
Z
7.8 8.2
X
26
Example

• For each of the following three populations,
  indicate what the sampling distribution for
  samples of 25 would consist of.
• Travel expense vouchers for a university in an
  academic year
• Absentee records (days absent per year) in 2002
  for employees in a large manufacturing company
• Yearly sales (in gallons) of unleaded gasoline at
  service stations located in a particular parish

27
Example

• The diameter of Ping-Pong balls manufactured at a
  large factory is expected to be approximately
  normally distributed with a mean of 1.30 inches
  and a standard deviation of 0.04 inch. What is
  the probability that a randomly selected
  Ping-Pong ball will have a diameter
• Less than 1.28 inches
• Between 1.31 and 1.33 inches
• Between what two values (symmetrically
  distributed) will 60 of the Ping-Pong balls fall
  (in terms of diameter)

28
Example contd

• If many random samples of 16 Ping-Pong balls are
  selected,
• What will be the values of the population mean
  and standard error of the mean?
• What distribution will the sample mean follow?
• What proportion of the sample means will be less
  than 1.28 inches?
• What proportion of the sample means will be
  between 1.31 and 1.33 inches?
• Between what two values symmetrically distributed
  around the mean will 60 of the sample means be?
• Compare the answers of (1) with (6) and (2) with
  (7). Discuss.
• Explain the difference in the results of (3) and
  (8).

29
Sampling Distribution of the Proportion
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
30
Population Proportions

• p the proportion of the population having
• some characteristic
• Sample proportion ( p ) provides an estimate
• of
  p
• 0 p 1
• p has a binomial distribution
• (assuming sampling with replacement from a
  finite population or without replacement from an
  infinite population)

31
Sampling Distribution of p

• Approximated by anormal distribution if
• Where and

Sampling Distribution
P( ps)
.3 .2 .1 0
p
0 . 2 .4 .6 8 1
(where p population proportion)
32
Z-Value for Proportions
Standardize p to a Z value with the formula
33
Example

• If the true proportion of voters who support
  Proposition A is p 0.4, what is the
  probability that a sample of size 200 yields a
  sample proportion between 0.40 and 0.45?

• i.e. if p 0.4 and n 200, what is
• P(0.40 p 0.45) ?

34
Example
(continued)

• if p 0.4 and n 200, what is
• P(0.40 p 0.45) ?

Find
Convert to standard normal
35
Example
(continued)

• if p 0.4 and n 200, what is
• P(0.40 p 0.45) ?

Use standard normal table P(0 Z 1.44)
0.4251
Standardized Normal Distribution
Sampling Distribution
0.4251
Standardize
0.45
1.44
0.40
0
Z
p
36
Example

• A recent survey has indicated that 20 of
  fine-dining restaurants have instituted policies
  restricting the use of cell phones. If random
  samples of 100 fine-dining restaurants are
  selected ,
• What proportion of samples are likely to have
  between 15 and 25 that have established
  policies restricting use?
• Within what symmetrical limits of the population
  percentage will 90 of the sample percentages
  fall? 95?

37
Sampling Distributions Means Proportions -
Review

• Means

• Proportions

38
Difference Between 2 Means or Proportions

• In some cases we may be interested in examining
  the sampling distribution of the difference
  between two means or two proportions
• The main assumption is that the samples are taken
  from large populations

39
Difference Between 2 Means or Proportions

• The sampling distribution of the difference
  between means can be thought of as the
  distribution that would result if we repeated the
  following three steps over and over again
• Sample n1 scores from Population 1and n2 scores
  from Population 2,
• Compute the means of the two samples
• Compute the difference between means
  . The distribution of the differences between
  means is the sampling distribution of the
  difference between means.

40
Difference 2 Means 2 Proportions

• Means

• Proportions

41
Sampling Distributions - Example I

• A statistics entrance examination consists of two
  sections, I II. The maximum mark that can be
  achieved on each section is 50. The population
  mean and standard deviation for both sections are
  
• Mean Std. Dev
• Section I 36 5.5
• Section II 42 4.5
• If a random sample of 100 candidates marks from
  Section I and 50 candidates marks from Section
  II are taken, find the probability that the mean
  mark obtained from Section I will be 5 marks or
  more, less than the mean mark from Section II.

42
Sampling Distributions - Example II

• It is estimated that 65 of the people who live
  in Portmore are comfortable in their homes. The
  corresponding figure for the people of Duhaney
  Park is 70. A random sample of 50 is taken from
  each of these two communities. Determine the
  probability that the sample suggests that the
  proportion in Portmore who are comfortable with
  their homes is greater than that of Duhaney Park.