Sampling Distributions

Sampling Distributions1IntroductionDiscrete distributionsBinomialPoissonHypergeometricContinuous distributionsNormalWe knew and used it to determine P(X)Using sample data to project to the population inferential statisticsSo far, weve looked at several discrete and one continuous probability distribution. In all of these, we emphasized looking at the entire population, but in the real world, we often have to rely on sampling data.

When we use sample data to make on estimate and project it on the entire population, we are making use of inferential statistics.

For starters, well continue to know the population mean, but well use the mean of the samples taken from the population to make probability statements. Later, well lose access to the population mean and rely solely on our sample data.2PreviewConsider the sample mean as a random variableDifferent values for sample meanOwn mean and s.d.Probability distribution of sample meansSample distribution of the mean

If we take a large number of samples of size n from the same population, we would end up with a great many different values for the sample mean. The resulting collection of sample means can then be viewed as a new random variable with its own mean and standard deviation. The probability distribution of these means is called the sampling distribution of the mean.For any specified population, the sampling distribution of the mean is defined as the probability distribution of the sample means for all possible samples of that particular size.

3The PopulationPersonxBill1Carl1Denise3Ed5 = (1+1+3+5)/4 = 2.5Each of four members of the population has a given number of bottles of cola in his or her refrigerator. Draw the probability distribution on the board.4All possible samples of n=2SampleMean of sampleP(sample)Bill, Carl(1+1)/2 = 1.01/6Bill, Denise(1+3)/2 = 2.01/6Bill, Ed(1+5)/2 = 3.01/6Carl, Denise(1+3)/2 = 2.01/6Carl, Ed(1+5)/2 = 3.01/6Denise, Ed(3+5)/2 = 4.01/6The sampling distribution of a statistic is the population of all possible values of that statistic. When a population is small, like this, we can compute the entire population. But it doesnt take long for it to get very big. Then we have to use a computer to demonstrate this principle.

Draw the distribution on the board.5Sampling Distribution of the MeanShow, also, that the sampling distribution of the is normally distributed although the original distribution was highly skewed (very small distribution size). Draw the distribution.

6Sampling Distribution of the MeanThe sampling distribution of the mean will always have the same mean as the original population.The standard deviation of the sampling distribution of the mean is referred to as the standard error of the mean.Standard error of the mean.Variance = Sum of (x-bar mu)^2 x probability Std error = square of variance7Sampling Distribution of the MeanIf the original population is distributed normally, the sampling population will also be normal.If the original population is not normal, the sampling distribution will approximate normal.Use Minitab to demonstrate. Then go to the online model.

8Sampling Distribution of the ProportionThe sample proportion can also be a random variable, which we will discuss next week.9Population Normally Distributed10Sampling Distribution of the MeanGiven the following probability distribution for an infinite population with the discrete RV, x:

x12P(x)0.50.5Determine the mean and standard deviation of x.For the sample size n=2, determine the mean for each simple random sample from the population.For each sample we just identified, what is the probability this sample will be selected?Combining the results we just got, describe the sampling distribution of the mean. Do this again for n=3. What effect does the change in sample size have on the mean and the standard error of the mean?11Effect of Sample SizeThe average annual hours flown by general aviation aircraft = 130. Assume these hours are normally distributed.Standard deviation = 30 hours.12Population

13Sampling Distribution, n=9

14Sampling Distribution, n=36

15PracticeA random variable is normally distributed with = $1500 and = $100. Determine the standard error of the mean for simple random samples with the following sample sizes:n=16n=100n=400n=1000

16Conversion to Standard Normal17Z-score ExampleAirplanes with = 130 and = 30For a simple random sample of 36 aircraft, what is the probability the average flight time for the aircraft in the sample was 138 hours?18Picking up where we left offA crane is operated by 4 electrical motors working together. For the crane to work properly, the 4 motors must generate 380 hp.Each motor produces an average of 100 hp with a standard deviation of 10 hp. Whats the probability that the crane wont work?Figure standard error of mean = 5Z = 380/4 = P x bar < 95 = P z 300 as listed on their label.The true mean of the population is 295, the is 12, and the population is normally distributed.What is the probability the load will be rejected?21PracticeThe average length of a hospital stay is 5.7 days. Assuming a of 2.5 days and a random sample of 50 patients, what is the probability the average stay for the sample will be 6.5 days?If the sample had been 8 instead of 50, what further assumption must we make in order to solve this problem?22ExampleWhat is the probability that a single, random reading will show this patient with BP above 150?What is the probability that the sample mean from 5 readings will show this patient with BP above 150?How many samples must we take before the probability of the sample mean being less than 150 is .01?23Proportions

24Proportions25Properties of the Sample ProportionProportions are actually binomial distributions

26Properties of the Sample Proportion27ExampleThe expected value of the sampling distribution of the proportion is p=.8510. This, along with the sample size of n=200 is used to determine the standard error of the sampling distribution and the z-score corresponding to a sample proportion of p=.8.Standard error = .0252Z = -2.02Probability = .021728Practice42.6% of all purchasing agents in the U.S. work force are women. In a random sample of 200 purchasing agents, 70 are women.What is the population proportion?What is the sample proportion?What is the standard error of the sample proportion?If we took another random sample, whats the probability wed get a proportion at least as large (.35) as the one we got here?Population proportion = .426Sample proportion = .35Standard error = .035Last part = .98529And finallyFinite PopulationsWhen we sample a relatively large part of the population, the sampling error is going to become very small. For example, suppose we sampled 900 out of a group of 1000. Wed expect to have almost zero sampling error. The effect becomes noticeable at 5%.30ExampleOf the 629 imported cars sold in Enterprise last year, 117 were Toyotas.A random sample of 300 imported cars is conducted.What is the probability that at least 15% of the vehicles in the sample will be Toyotas?Standard error= 0.0163Z = -2.21Answer = .986431