Chapter 7 Sampling and Sampling Distributions

1
Chapter 7Sampling and Sampling Distributions

• Simple Random Sampling
• Point Estimation
• Introduction to Sampling Distributions
• Sampling Distribution of
• Sampling Distribution of p

n 100
n 30
2
Statistical Inference

• The purpose of statistical inference is to obtain
  information about a population from information
  contained in a sample.
• A population is the set of all the elements of
  interest.
• A sample is a subset of the population.
• The sample results provide only estimates of the
  values of the population characteristics.
• A parameter is a numerical characteristic of a
  population.
• With proper sampling methods, the sample results
  will provide good estimates of the population
  characteristics.

3
Simple Random Sampling

• A simple random sample from a finite population
  of size N is a sample selected such that each
  possible sample of size n has the same
  probability of being selected.
• Replacing each sampled element before selecting
  subsequent elements is called sampling with
  replacement.
• Sampling without replacement is the procedure
  used most often.
• In large sampling projects, computer-generated
  random numbers are often used to automate the
  sample selection process.

• Finite Population

4
Simple Random Sampling

• A simple random sample from an infinite
  population is a sample selected such that the
  following conditions are satisfied.
• Each element selected comes from the same
  population.
• Each element is selected independently.
• The population is usually considered infinite if
  it involves an ongoing process that makes listing
  or counting every element impossible.
• The random number selection procedure cannot be
  used for infinite populations.

• Infinite Population

5
Point Estimation

• In point estimation we use the data from the
  sample to compute a value of a sample statistic
  that serves as an estimate of a population
  parameter.
• We refer to as the point estimator of the
  population mean ?.
• s is the point estimator of the population
  standard deviation ?.
• p is the point estimator of the population
  proportion ?.

6
Sampling Error

• The absolute difference between an unbiased point
  estimate and the corresponding population
  parameter is called the sampling error.
• Sampling error is the result of using a subset of
  the population (the sample), and not the entire
  population to develop estimates.
• The sampling errors are
• for sample mean
• for sample standard deviation
• for sample proportion

7
Example St. Edwards

• St. Edwards University receives 900
  applications
• annually from prospective students. The
  application
• forms contain a variety of information including
  the
• individuals scholastic aptitude test (SAT) score
  and
• whether or not the individual is an in-state
  resident.
• The director of admissions would like to know,
  at
• least roughly, the following information
• the average SAT score for the applicants, and
• the proportion of applicants that are in-state
  residents.
• We will now look at two alternatives for
  obtaining
• the desired information.

8
Example St. Andrews

• Alternative 1 Take a Census of the 900
  Applicants
• SAT Scores
• Population Mean
• Population Standard Deviation
• In-State Residents
• Population Proportion

9
Example St. Edwards

• Alternative 2 Take a Sample of 30 Applicants
• Excel can be used to select a simple random
  sample without replacement.
• The process is based on random numbers generated
  by Excels RAND function.
• RAND function generates numbers in the interval
  from 0 to 1.
• Any number in the interval is equally likely.
• The numbers are actually values of a uniformly
  distributed random variable.

10
Example St. Edwards

• Using Excel to Select a Simple Random Sample
• 900 random numbers are generated, one for each
  applicant in the population.
• Then we choose the 30 applicants corresponding to
  the 30 smallest random numbers as our sample.
• Each of the 900 applicants have the same
  probability of being included.

11
Using Excel to Selecta Simple Random Sample

• Formula Worksheet

Note Rows 10-901 are not shown.
12
Using Excel to Selecta Simple Random Sample

• Value Worksheet

Note Rows 10-901 are not shown.
13
Using Excel to Selecta Simple Random Sample

• Put Random Numbers in Ascending Order
• Step 1 Select cells A2A901
• Step 2 Select the Data pull-down menu
• Step 3 Choose the Sort option
• Step 4 When the Sort dialog box appears
• Choose Random Numbers
• in the Sort by text box
• Choose Ascending
• Click OK

14
Using Excel to Selecta Simple Random Sample

• Value Worksheet (Sorted)

Note Rows 10-901 are not shown.
15
Example St. Andrews

• Point Estimates
• as Point Estimator of ?
• s as Point Estimator of ?
• p as Point Estimator of ?
• Note Different random numbers would have
• identified a different sample which would have
  resulted in different point estimates.

16
Example St. Andrews, Sampling Errors
17
Sampling Distribution of

• The sampling distribution of is the
  probability distribution of all possible values
  of the sample
• mean .

• If there are 200 students in this room (N 200),
  and I want to select a sample of 30 students (n
  30), how many different samples of n 30 are
  possible?

18
Sampling Distribution of

• The sampling distribution of is the
  probability distribution of all possible values
  of the sample
• mean .

19
Sampling Distribution of

• The sampling distribution of is the
  probability distribution of all possible values
  of the sample
• mean .
• Expected Value of
• E( ) ?
• where
• ? the population mean

20
Sampling Distribution of

21
Sampling Distribution of

• Standard Deviation of
• Finite Population Infinite
  Population
• A finite population is treated as being infinite
  if n/N lt .05.
• is the finite correction
  factor.
• is referred to as the standard error of the
  mean.

22
Sampling Distribution of

• If we use a large simple random sample, the
  central limit theorem enables us to conclude that
  the sampling distribution of can be
  approximated by a normal probability
  distribution.

23
Central Limit Theorem

• 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

24
Central Limit Theorem
the sampling distribution becomes almost normal
regardless of shape of population
As the sample size gets large enough
n?
25
How Large is Large Enough?

• For most distributions, n 30 will give a
  sampling distribution that is nearly normal
• For fairly symmetric distributions, n 15 is
  sufficient
• For normal population distributions, the sampling
  distribution of the mean is always normally
  distributed
• When the simple random sample is small (n lt 30),
  the sampling distribution of can be
  considered normal only if we assume the
  population has a normal probability distribution.

26
Example St. Andrews

• Sampling Distribution of for the SAT Scores

27
Example St. Andrews

• Sampling Distribution of for the SAT Scores
• What is the probability that a simple random
  sample of 30 applicants will provide an estimate
  of the population mean SAT score that is within
  plus or minus 10 of the actual population mean ?
  ?
• In other words, what is the probability that
  will be between 980 and 1000?

28
Example St. Andrews

• Sampling Distribution of for the SAT Scores

Sampling distribution of
1000
980
990
Using the standard normal probability table
z1000 10/14.6 .68, and z980 -10/14.6
-.68, we have area .7517 - .2483 .5034
29
Now You Try. Pg. 310, 7-36
30
Working With Proportions

• ? the proportion of the population having some
  characteristic.
• Where
• ? Population proportion
• X Number of items in the population
  with the characteristic of interest
• N Population size

31
Working With Proportions

• Sample Proportion
• Where
• p Sample proportion
• x Number of items in the sample
  with the
  characteristic of interest
• n Sample size

32
Sampling Distribution of p

• The sampling distribution of p is the probability
  distribution of all possible values of the sample
  proportion
• Expected Value of p
• where
• ? the population proportion

33
Sampling Distribution of p

• Standard Deviation of p
• Finite Population Infinite Population
• is referred to as the standard error of the
  proportion.

• A finite population is treated as being infinite
  if n/N lt .05.

34
Sampling Distribution of p

• The sampling distribution of p can be
  approximated by a normal probability distribution
  whenever the sample size is large.
• Two conditions for a large sample size

35
(No Transcript)
36
Example St. Andrews

• Sampling Distribution of p for In-State Residents
• The normal probability distribution is an
  acceptable approximation since n? 30(.72)
  21.6 gt 5 and
• n(1 - ?) 30(.28) 8.4 gt 5.

37
Example St. Andrews

• Sampling Distribution of p for In-State Residents
• What is the probability that a simple random
  sample of 30 applicants will provide an estimate
  of the population proportion of applicants who
  are in-state residents that is within plus or
  minus .05 of the actual population proportion?
• In other words, what is the probability that p
  
• will be between .67 and .77?

38
Example St. Andrews

• Sampling Distribution of p for In-State Residents

Sampling distribution of p
0.77
0.67
0.72
z.77 .05/.082 .61, and z.67 -.05/.082
-.61 Therefore, the area .7291 - .2709
.4582. The probability is .4582 that the sample
proportion will be within /-.05 of the actual
population proportion.
39
Sampling and Sampling Distributions

• The mean weight of the students at UCF is 150
  pounds with a standard deviation of 15, and
• the proportion of the student body that exercises
  at least 3 times per week is .65

• If enrollment is 48,000 students and I select a
  random sample of 30 students,
• What is the probability that the sample mean will
  be within 5 pounds of the population mean?
• What is the probability that the sample
  proportion will be within .05 of the population
  proportion?

40
Example Class Fitness

• Sampling Distribution of for the class
  weights

41
Example Class Fitness

• Sampling Distribution of for student weights

155
145
150

• Using the standard normal probability table with
  z 5/2.74 1.82, we have area .9656 -
  .0344 .9312

42
Example Class Fitness

• Sampling Distribution of for students who
  exercise

43
Example Student Fitness

• Sampling Distribution of for students who
  exercise
• For z .05/.087 .57, the area .7157 - .2843
  .4314.
• The probability is .4314 that the sample
  proportion will be within /-.05 of the actual
  population proportion.

0.70
0.60
0.65
44
Now You Try Page 320, 7-55
45
End of Chapter 7