LOUCKS
St. ASW0324360681_amzn
JOHN Edward's University
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Slide 1
Chapter 7, Part A Sampling and Sampling Distributions
Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of x
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Slide 2
Statistical Inference
The purpose of statistical inference is to obtain information about a population from information contained in a sample. An element is the entity on which data are collected. A population is the set of all the elements of interest. A sample is a subset of the population.
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Slide 3
Statistical Inference
The sample results provide only estimates of the values of the population characteristics. With proper sampling methods, the sample results can provide "good" estimates of the population characteristics. A parameter is a numerical characteristic of a population.
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Slide 4
Statistical Inference
A sampled population is the population from which the sample is drawn. A frame is a list of the elements that the sample will be selected from.
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Slide 5
Statistical Inference
When making inferences about a population based on a sample, it is important to have a close correspondence between the sampled population and the target population. The target population is the population we want to make inferences about.
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Slide 6
Simple Random Sampling From a Finite Population
s
Finite populations are often defined by lists such as: Organization membership roster Credit card account numbers Inventory product numbers
s A simple random sample of size n 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.
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Slide 7
Simple Random Sampling From a Finite Population
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, computergenerated random numbers are often used to automate the sample selection process.
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Slide 8
Simple Random Sampling From an Infinite Population
s s
Sometimes the sampled population is such that a frame cannot be constructed. One such situation is sampling from an ongoing process where the conceptual population is infinite.
s
Another situation is sampling from a possibly very large population where it is not possible, or perhaps feasible, to identify all the elements in the population. s In these situations the random number selection procedure cannot be used.
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Slide 9
Simple Random Sampling From an Infinite Population
In the case of infinite populations, it is impossible to obtain a list of all elements in the population.
s 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.
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Slide 10
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.
x 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 p.
Slide 11
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Example: St. Andrew's
St. Andrew's College receives 900 applications annually from prospective students. The application form contains a variety of information including the individual's scholastic aptitude test (SAT) score and whether or not the individual desires oncampus housing.
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Slide 12
Example: St. Andrew's
The director of admissions would like to know the following information: the average SAT score for the 900 applicants, and the proportion of applicants that want to live on campus.
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Slide 13
Example: St. Andrew's
We will now look at two alternatives for obtaining the desired information. s Conducting a census of the entire 900 applicants
s
Selecting a sample of 30 applicants, using Excel
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Slide 14
Conducting a Census
s
If the relevant data for the entire 900 applicants were in the college's database, the population parameters of interest could be calculated using the formulas presented in Chapter 3. We will assume for the moment that conducting a census is practical in this example.
s
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Slide 15
Conducting a Census
s
s
Population Mean SAT Score x i = 990 = 900 Population Standard Deviation for SAT Score
=
s
( x i - )2 900
= 80
Population Proportion Wanting OnCampus Housing
648 p= = .72 900
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Slide 16
Simple Random Sampling
Now suppose that the necessary data on the current year's applicants were not yet entered in the college's database. Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for a meeting taking place in a few hours. She decides a sample of 30 applicants will be used. The applicants were numbered, from 1 to 900, as their applications arrived.
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Slide 17
Using a Random Number Table
s
Simple Random Sampling:
Taking a Sample of 30 Applicants
Because the finite population has 900 elements, we
will need 3digit random numbers to randomly select applicants numbered from 1 to 900.
We will use the last three digits of the 5digit
random numbers in the third column of the textbook's random number table, and continue into the fourth column as needed.
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Slide 18
Using a Random Number Table
s
Simple Random Sampling:
Taking a Sample of 30 Applicants
The numbers we draw will be the numbers of the
applicants we will sample unless the random number is greater than 900 or the random number has already been used. We will continue to draw random numbers until we have selected 30 applicants for our sample.
(We will go through all of column 3 and part of
column 4 of the random number table, encountering in the process five numbers greater than 900 and one duplicate, 835.)
Slide 19
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Using a Random Number Table
s
Simple Random Sampling:
Use of Random Numbers for Sampling 3Digit Applicant Random Number Included in Sample 744 No. 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds 900 190 No. 190 836 No. 836 . . . and so on
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Slide 20
Using a Random Number Table
s
Simple Random Sampling:
Sample Data Random SAT Live On Score Campus No. Number Applicant 1 744 Conrad Harris 1025 Yes 2 436 Enrique Romero 950 Yes 3 865 Fabian Avante No 1090 4 790 Lucila Cruz 1120 Yes 5 835 Chan Chiang 930 No . . . . . . . . . . 30 498 Emily Morse 1010 No
Slide 21
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Simple Random Sampling:
Using a Computer
s
Taking a Sample of 30 Applicants Computers can be used to generate random numbers for selecting random samples. For example, Excel's function = RANDBETWEEN(1,900) can be used to generate random numbers between 1 and 900. Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample.
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Slide 22
Point Estimation
s as Point Estimator of x
x=
xi n
=
29, 910 = 997 30
s s as Point Estimator of
s=
( x i - x )2 n-1
=
163, 996 = 75.2 29
s as Point Estimator of p p
p = 20 30 = .68
Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.
2009 Thomson SouthWestern. All Rights Reserved Slide 23
Summary of Point Estimates Obtained from a Simple Random Sample
Population Parameter Parameter Value 990 80 Point Estimator Point Estimate 997 75.2
= Population mean SAT score = Population std. deviation for SAT score
p = Population pro portion wanting campus housing
x = Sample mean
SAT score s = Sample std. deviation for SAT score
.72
p = Sample pro portion wanting campus housing
.68
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Slide 24
x Sampling Distribution of
s
Process of Statistical Inference Population with mean = ? A simple random sample of n elements is selected from the population.
x The value of is used to make inferences about the value of .
The sample data provide a value for x the sample mean .
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Slide 25
x Sampling Distribution of
x The sampling distribution of is the probability distribution of all possible values of the sample x mean .
Expected Value of
x x E( ) =
where: = the population mean
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Slide 26
x Sampling Distribution of
Standard Deviation of x
x = n
N -n x = ( ) n N -1
Standard Deviation of x for a Finite Population
A finite population is treated as being infinite if n/N < .05. ( N - n ) / ( N - 1) is the finite correction factor.
x is referred to as the standard error of the mean.
2009 Thomson SouthWestern. All Rights Reserved Slide 27
Form of the Sampling Distribution of x
When the population has a normal distribution, the x sampling distribution of is normally distributed for any sample size. In most applications, the sampling distribution of x can be approximated by a normal distribution whenever the sample is size 30 or more. In cases where the population is highly skewed or outliers are present, samples of size 50 may be needed.
Slide 28
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x Sampling Distribution of for SAT Scores
Sampling Distribution x of
x =
80 = = 14.6 n 30
E( x ) = 990
x
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Slide 29
x Sampling Distribution of for 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 +/- 10 of the actual population mean ? x In other words, what is the probability that will be between 980 and 1000?
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Slide 30
x Sampling Distribution of for SAT Scores
Step 1: Calculate the zvalue at the upper endpoint of the interval. z = (1000 - 990)/14.6= .68 Step 2: Find the area under the curve to the left of the upper endpoint. P(z < .68) = .7517
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Slide 31
x Sampling Distribution of for SAT Scores
Cumulative Probabilities for the Standard Normal Distribution
z . .5 .6 .7 .8 .9 . .00 . .01 . .02 . .03 . .04 . .05 . .06 . .07 . .08 . .09 .
.6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 . . . . . . . . . .
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Slide 32
x Sampling Distribution of for SAT Scores
Sampling Distribution x of Area = .7517
x = 14.6
990 1000
x
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Slide 33
x Sampling Distribution of for SAT Scores
Step 3: Calculate the zvalue at the lower endpoint of the interval. z = (980 - 990)/14.6= .68 Step 4: Find the area under the curve to the left of the lower endpoint. P(z < .68) = .2483
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Slide 34
x Sampling Distribution of for SAT Scores
Sampling Distribution x of Area = .2483
x = 14.6
980 990
x
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Slide 35
x Sampling Distribution of for SAT Scores
Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. P(.68 < z < .68) = P(z < .68) - P(z < .68) = .7517 - .2483 = .5034 The probability that the sample mean SAT score will be between 980 and 1000 is:
x P(980 < < 1000) = .5034
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Slide 36
x Sampling Distribution of for SAT Scores
Sampling Distribution x of
x = 14.6
Area = .5034
980 990 1000
x
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Slide 37
Relationship Between the Sample Size x and the Sampling Distribution of
Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. E( ) = regardless of the sample size. In our x x example, E( ) remains at 990. Whenever the sample size is increased, the standard x error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the mean is decreased to:
80 x = = = 8.0 n 100
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Slide 38
Relationship Between the Sample Size x and the Sampling Distribution of
With n = 100, x = 8 With n = 30, x = 14.6
E( x ) = 990
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x
Slide 39
Relationship Between the Sample Size x and the Sampling Distribution of
Recall that when n = 30, P(980 < < 1000) = .5034. x We follow the same steps to solve for P(980 < x 1000) < when n = 100 as we showed earlier when n = 30. Now, with n = 100, P(980 < < 1000) = .7888. x Because the sampling distribution with n = 100 has a x smaller standard error, the values of have less variability and tend to be closer to the population x mean than the values of with n = 30.
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Slide 40
Relationship Between the Sample Size x and the Sampling Distribution of
Sampling Distribution x of
x = 8
Area = .7888
980 990 1000
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x
Slide 41
End of Chapter 7, Part A
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Slide 42