Unformatted text preview: Chapter 7 Sampling and Sampling Distributions
s s s s s Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of x Sampling Distribution of p n = 100 n = 30 Slide 1 Statistical Inference
s s s s s s 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. Slide 2 Simple Random Sampling
s Finite Population • 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, computergenerated random numbers are often used to automate the sample selection process. Slide 3 Simple Random Sampling
s Infinite Population • 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. Slide 4 Point Estimation
s s s s 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 µ. s is the point estimator of the population standard deviation σ . p is the point estimator of the population proportion p. Slide 5 Sampling Error
s s s 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:  x − µ  for sample mean  s σ  for sample standard deviation  p − p  for sample proportion Slide 6 Example: St. Andrew’s
St. Andrew’s University receives 900 applications annually from prospective students. The application forms contain a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires oncampus housing. Slide 7 Example: St. Andrew’s
The director of admissions would like to know the following information: • the average SAT score for the applicants, and • the proportion of applicants that want to live on campus. We will now look at three alternatives for obtaining the desired information. • Conducting a census of the entire 900 applicants • Selecting a sample of 30 applicants, using a random number table • Selecting a sample of 30 applicants, using computer generated random numbers Slide 8 Example: St. Andrew’s
s Taking a Census of the 900 Applicants • SAT Scores • Population Mean ∑x µ= i 900 = 990 • Population Standard Deviation
σ= ∑ (x i − µ )2 • 900 = 80 Applicants Wanting OnCampus Housing • Population Proportion
p= 648 = .72 900 Slide 9 Example: St. Andrew’s
s Take a Sample of 30Applicants Using a Random Number Table Since 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 a random number table. 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. Slide 10 Example: St. Andrew’s
s Use of Random Numbers for Sampling 3Digit Random Number 744 436 865 790 835 902 190 436 etc. Applicant Included in Sample No. 744 No. 436 No. 865 No. 790 No. 835 Number exceeds 900 No. 190 Number already used etc. Slide 11 Example: St. Andrew’s
s Sample Data Random No. Number Applicant SAT Score OnCampus 1 744 Connie Reyman 1025 Yes 2 436 William Fox 950 Yes 3 865 Fabian Avante 1090 No 4 790 Eric Paxton 1120 Yes 5 835 Winona Wheeler 1015 No . . . . . 30 685 Kevin Cossack 965 No Slide 12 Example: St. Andrew’s
s Take a Sample of 30 Applicants Using Computer Generated Random Numbers • Excel provides a function for generating random numbers in its worksheet. • 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. Slide 13 Using Excel to Select a Simple Random Sample
s Formula Worksheet
A 1 2 3 4 5 6 7 8 9 Applicant Number 1 2 3 4 5 6 7 8 B SAT S core 1008 1025 952 1090 1127 1015 965 1161 C OnCampus Housing Yes No Yes Yes Yes No Yes No D Random Number =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() Note: Rows 10901 are not shown. Slide 14 Using Excel to Select a Simple Random Sample
s Value Worksheet
A 1 2 3 4 5 6 7 8 9 Applicant Number 1 2 3 4 5 6 7 8 B SAT S core 1008 1025 952 1090 1127 1015 965 1161 C OnCampus Housing Yes No Yes Yes Yes No Yes No D Random Number 0.41327 0.79514 0.66237 0.00234 0.71205 0.18037 0.71607 0.90512 Note: Rows 10901 are not shown. Slide 15 Using Excel to Select a Simple Random Sample
s Value Worksheet (Sorted)
A 1 2 3 4 5 6 7 8 9 Applicant Number 12 773 408 58 116 185 510 394 B SAT Score 1107 1043 991 1008 1127 982 1163 1008 C OnCampus Housing No Yes Yes No Yes Yes Yes No D Random Number 0.00027 0.00192 0.00303 0.00481 0.00538 0.00583 0.00649 0.00667 Note: Rows 10901 are not shown. Slide 16 Example: St. Andrew’s
s Point Estimates x • as Point Estimator of µ ∑x x= • s as Point Estimator of σ
s= ( xi x )2 ∑ 29 29,910 = = 997 30 30
i = p • as Point Estimator of p
s 163,996 = 75.2 29 p = 20 30 = .68 Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. Slide 17 Sampling Distribution of x
s Process of Statistical Inference Population with mean µ = ? 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 µ . The sample data provide a value for x the sample mean . Slide 18 Sampling Distribution of x
s s x The sampling distribution of is the probability distribution of all possible values of the sample x mean . Expected Value of x x E( ) = µ where: µ = the population mean Slide 19 Sampling Distribution of x
s Standard Deviation of Finite Population x Infinite Population σ N −n σx = ( ) n N −1 σ σx = n • • • A finite population is treated as being infinite if n/N < .05. ( N − n ) / ( N − 1) is the finite correction factor. σ x is referred to as the standard error of the mean. Slide 20 Sampling Distribution of x
s If we use a large (n > 30) simple random sample, the central limit theorem enables us to conclude that the x sampling distribution of can be approximated by a normal probability distribution. When the simple random sample is small (n < 30), the x sampling distribution of can be considered normal only if we assume the population has a normal probability distribution. s Slide 21 Example: St. Andrew’s
s x Sampling Distribution of for the SAT Scores σ 80 σx = = = 14.6 n 30 E ( x ) = µ = 990 x Slide 22 Example: St. Andrew’s
s x 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 µ ? ? x In other words, what is the probability that will be between 980 and 1000? Slide 23 Example: St. Andrew’s
s x Sampling Distribution of for the SAT Scores
Sampling distribution of x Area = .2518 Area = .2518 980 990 1000 x Using the standard normal probability table with z = 10/14.6= .68, we have area = (.2518)(2) = .5036 Slide 24 Sampling Distribution of p
s p The sampling distribution of is the probability distribution of all possible values of the sample proportion p Expected Value of p where: s E ( p) = p p = the population proportion Slide 25 Sampling Distribution of p
s Standard Deviation of Finite Population p Infinite Population σp = p (1 − p ) N − n n N −1 σp = p (1 − p ) n • σ p is referred to as the standard error of the proportion. Slide 26 Example: St. Andrew’s
s p Sampling Distribution of for InState Residents .72(1 − .72) σp = = .082 30 E( p ) = .72 The normal probability distribution is an acceptable approximation since np = 30(.72) = 21.6 > 5 and n(1 p) = 30(.28) = 8.4 > 5. Slide 27 Example: St. Andrew’s
s p Sampling Distribution of for InState Residents What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring oncampus housing that is within plus or minus .05 of the actual population proportion? p In other words, what is the probability that will be between .67 and .77? Slide 28 Example: St. Andrew’s
s p Sampling Distribution of for InState Residents
Sampling distribution of p Area = .2291 Area = .2291 0.67 0.72 0.77 p For z = .05/.082 = .61, the area = (.2291)(2) = .4582. The probability is .4582 that the sample proportion will be within +/.05 of the actual population proportion. Slide 29 ...
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 Spring '08
 Priya
 Simple random sample, St. Andrew

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