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 for sample standard deviation  p − p  for sample proportion Slide 6 Example: St. Andrew’s College
St. Andrew’s College 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 two alternatives for obtaining the desired information. • Conducting a census of the entire 900 applicants • Selecting a sample of 30 applicants, using computer generated random numbers in Excel Slide 8 Conducting a Census
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 Simple Random Sampling Using Excel
Take a Sample of 30 Applicants Using ComputerGenerated Random Numbers • 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. • Each of the 900 applicants have the same probability of being included.
s Slide 10 Point Estimation
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 11 Summary of Point Estimates Obtained from a Simple Random Sample
Population Parameter Parameter Value 990 80 Point Estimator Point Estimate 997 75.2 µ = Population mean SAT score σ = 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 Slide 12 Sampling Error s Sampling error for mean = s Sampling error for standard deviation = s Sampling error for proportion = Slide 13 Sampling Distribution of
s x 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 14 Sampling Distribution of s x x The sampling distribution of is the probability distribution of all possible values of the sample mean . x
Is the sample mean a random variable? Expected Value of x s s x E( ) = µ where: µ = the population mean Slide 15 Sampling Distribution of s x x Standard Deviation of (standard error of the mean) Finite Population 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 population correction factor (FPCF). σ x is referred to as the standard error of the mean. Slide 16 Example
s Example: Suppose a simple random sample of size 49 is selected from a population with σ=14. Find the value of the standard error of the mean in each of the following cases. a. The population size is infinite. a. a. The population size in N=500. The population size in N=4900. Slide 17 Sampling Distribution of s x Central Limit Theorem: If we use a large (n > 30) sample from a population (any distribution), then the sampling x distribution of can be approximated by a normal distribution. (See Figure 7.3, ASW p. 273 for examples.)
s s When the sample is small (n < 30), the sampling distribution x of can be considered normal only if the population has a normal distribution. If the original distribution of the population is normal, then the distribution of the sample mean is always normal, irrespective of the sample size. Slide 18 Sampling Distribution of s x x In order to characterize the distribution of , we need x E( ) = µ and, σ = σ
x n s x − E(x ) z= σ x The standard normal random variable is defined as Q. Why are we interested in the sampling distribution of x ? To calculate the probabilistic information about the difference between the sample mean and the population mean. Slide 19 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 20 Example: St. Andrew’s
s x Sampling Distribution of for the SAT Scores Q. 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 µ ? ? x In other words, what is the probability that will be between 980 and 1000? Slide 21 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 22 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( ) = µ regardless of the sample size. In our x x example, E( ) remains at 990. Whenever the sample size is increased, the standard σx 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:
σ 80 σx = = = 8.0 n 100 Slide 23 Relationship Between the Sample Size x and the Sampling Distribution of With n = 100, σ x = 8 With n = 30, σ x = 14.6 E( x ) = 990 x Slide 24 Relationship Between the Sample Size x and the Sampling Distribution of Recall that when n = 30, P(980 < < 1000) = .5036. 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 Because the sampling distribution with n = 100 has a x smaller standard error, the values of have less smaller standard error, the values of have less variability and tend to be closer to the population variability and tend to be closer to the population x mean than the values of with n = 30. The sampling mean than the values of with n = 30. The sampling x distribution of distribution of becomes narrower and more peaked. Slide 25 Sampling Distribution of p
Let X be the number of successes in n trials. Then X is Binomial (n, p). s Let denote the sample proportion of successes. Then p
s p=
s X n p The sampling distribution of is the probability distribution of all possible values of the sample proportion p
Expected Value of s p: E ( p) = p where, p = the population proportion Slide 26 Sampling Distribution of p
s Standard Deviation of p (standard error of the proportion) Finite Population
p (1 − p ) N − n n N −1 Infinite Population σp = σp = p (1 − p ) n • • A finite population is treated as being infinite if n/N < . 05. is referred to as the standard error of the proportion. σp Slide 27 Form of the Sampling Distribution of p p The sampling distribution of can be approximated by a normal distribution whenever the sample size is large. The sample size is considered large whenever both of the following two conditions are satisfied: np > 5 and n(1 – p) > 5 Slide 28 Form of the Sampling Distribution of p For values of p near .50, sample sizes as small as 10 permit a normal approximation. With very small (approaching 0) or very large (approaching 1) values of p, much larger samples are needed. Slide 29 Example: St. Andrew’s
s Sampling Distribution of p Q. 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 +/ .05 of the actual population proportion? In other words, what is the probability that p will be between .67 and .77? Slide 30 Example: St. Andrew’s
s Sampling Distribution of p .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 31 Example: St. Andrew’s
s Sampling Distribution of p 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 32 ...
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 Summer '08
 Priya
 Simple random sample, St. Andrew

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