1. Chapter 8 Sampling Variability and Sampling Distributions

2. JellyBlubbers:Episode 4 A New Curve Well, JellyBlubbers have taken control of my backyard, but a solution to the JellyBlubber invasion is in the works! I have determined that Jelly aggression is the result of the skewed nature of the distribution of their lengths. Jellies require balance in all things and as long as they continue to focus on such a skewed distribution it upsets them. So, lets see if we can help the Jellies out!

3. JellyBlubbers:Episode 4 A New Curve 1.Use a random number generator (a table or your calculator) to generate a random sample of 20 JellyBlubbers. 2.Find the Sample Mean length of the set. 3.Graph the sample mean length on the dot- plot at the front of the room. 4.Repeat the process with a new random sample. If we wanted to do this for every possible sample of size 20, how many sample mean lengths would we have to find?

4. JellyBlubbers:Episode 4 A New Curve Does it appear that this new distribution the Jellies will focus on, the sampling distribution of the sample means, will allow us to achieve the peace and balance in my backyard? Why?

5. Statistic: Any quantity computed from values in a sample. Sampling variability: The observed value of a statistic depends on the particular sample selected from the population; Typically, it varies from sample to sample. BASIC TERMS

6. Sample This! Q: How many possible different samples of 5 m&ms are in a bag of 500 m&ms? A: Too many…Not really. There are 500 C 5 or 255,244,687,600 possible samples of 5 M&Ms. This is called the Population of Samples. 500 C 5

7. Sampling Distribution: The distribution of a statistic. If the population of samples is relatively small, a sampling distribution can be displayed in a table just like any other probability distribution! BASIC TERMS, cont.

8.  =  = MHS has the following senior football starters and their weights in pounds: Aaron-220, Brad-200, Chris-170, Derek-180, Eric-190, Frank-210, and George-160. Suppose this is the population we are interested in. The mean and standard deviation of this population are: Sampling Distribution of Sample Means: An Example…

9. To create a sampling distribution of sample means from every combination of two players, let’s create the population of samples. Create the sampling distribution as a probability distribution then create the dotplot for the original data and for the sample means. Compare them.

10. Results!!! Original data  = 190  = 20  Graph Sample means  = 190  = 12.91  Graph

11. Now create the population OF SAMPLES for samples of size 3! Create the sampling distribution and the dotplot.dotplot

12. What Do You Notice About the Sampling Distributions of Sample Means As the Sample Size Increases From the Parent Population? What type of distribution is the parent population? What is the mean (center) of the parent population?

13. What Do You Notice About the Sampling Distributions of Sample Means As the Sample Size Increases From the Parent Population? What is the standard deviation of the parent population? What shape (type of distribution) is the sampling distribution of sample means of sample size 2?

14. What Do You Notice About the Sampling Distributions of Sample Means As the Sample Size Increases From the Parent Population? What shape (type of distribution) is the sampling distribution of sample means of sample size 3? Remind me. What was the center (mean) value of the sampling distribution of sample means of sample size 2?

15. What Do You Notice About the Sampling Distributions of Sample Means As the Sample Size Increases From the Parent Population? What appears to be the center (mean) value of the sampling distribution of sample means of sample size 3? The standard deviation of the sampling distribution of sample means of sample size 2 is 13.2288 and for sample size 3 is 9.5665. How does this compare to the parent population standard deviation of 20?

16. Example 2… Consider a very large population that consists of the numbers 1, 2, 3, 4 and 5 generated in a manner that the probability of each of those values is 0.2 no matter what the previous selections were. This population could be described as the outcome associated with a spinner such as given below with the distribution next to it.

17. Example 2 If the sampling distribution for the means of samples of size two is analyzed, it looks like….. Sampling DistributionPopulation of Samples

18. Example 2 The original distribution and the sampling distribution of means of samples with n=2 are given below. 5432 1 54321 Original distribution Sampling distribution n = 2

19. Example 2 Sampling distributions for n=3 and n=4 were calculated and are illustrated below. 5432 1 Original distribution 5432 1 Sampling distribution n = 2 Sampling distribution n = 3 54321 Sampling distribution n = 4 5432 1

20. Simulations 4 3 2 Means (n=120) 432 Means (n=60) 432 Means (n=30) To illustrate the general behavior of samples of fixed size n, 10000 samples each of size 30, 60 and 120 were generated from this uniform distribution and the means calculated. Probability histograms were created for each of these (simulated) sampling distributions. Notice all three of these look to be essentially normally distributed. Also the mean of each is 3 and the variability decreases as the sample size increases.

21. Simulations Skewed distribution To further illustrate the general behavior of samples of fixed size n, 10000 samples each of size 4, 16 and 30 were generated from the positively skewed distribution pictured below. Notice that these sampling distributions all all skewed, but as n increased the sampling distributions became more symmetric and eventually appeared to be almost normally distributed.

22. Terminology

23. Properties of the Sampling Distribution of the Sample Mean. * *10% depending on your text of choice. Its just like the Binomial Distribution rule for sampling without replacement.

24. Rule 4: When n is sufficiently large, the sampling distribution of is approximately normally distributed, even when the population distribution is not itself normal. Central Limit Theorem Central Limit Theorem.

25. Illustrations of Sampling Distributions Symmetric normal-like population

26. Illustrations of Sampling Distributions Skewed population

27. More About the Central Limit Theorem. The Central Limit Theorem can safely be applied when n exceeds 30. If n is large or the population distribution is normal, the standardized variable, z, has (approximately) a standard normal (z) distribution.

28. Examples Example 1: Non-CLT problem The average number of detention hours assigned per offender at MHS is 5 hours with a standard deviation of 1.5 hours. If an offender is selected at random what is the probability he/she served no more than 7 hours of detention?  = 5,  = 1.5

29. CLT Same Setup: What is the probability that a random sample of 30 offenders will have served an average of at most 7 hours? Notice now we are talking about an average from a sample of items not an individual item. This is CLT.

30. Example A food company sells “18 ounce” boxes of cereal. Let x denote the actual amount of cereal in a box of cereal. Suppose that x is normally distributed with  = 18.03 ounces and  = 0.05. a)What proportion of the boxes will contain less than 18 ounces?

31. Example - Continued b)A case consists of 24 boxes of cereal. What is the probability that the mean amount of cereal (per box in a case) is less than 18 ounces? The central limit theorem states that the distribution of is normal so…

33. Some Proportion Distributions Where P = 0.2 0.2 n = 10 0.2 n = 50 0.2 n = 20 0.2 n = 100 Let be the proportion of successes in a random sample of size n from a population whose proportion of S’s (successes) is p. * * Or  depending on your textbook of choice.

34. Properties of the Sampling Distribution of Let be the proportion of successes in a random sample of size n from a population whose proportion of S’s (successes) is p. Denote the mean of by mu and the standard deviation by sigma. Then the following rules hold

35. Properties of the Sampling Distribution ofSampling Distribution Rule 3: Rule 1: Rule 2: When n is large and p is not too near 0 or 1, the sampling distribution of p-hat is approximately normal. (CLT for proportions.) Rule of thumb: If n·p  10 and n(1-p)  10 the distribution is approximately normal

36. Condition for Use The further the value of p is from 0.5, the larger n must be for a normal approximation to the sampling distribution of to be accurate. Rule of Thumb: Remember!!!!!! If both np  10 and n(1-p)  10, then it is safe to use a normal approximation.

37. Example If the true proportion of defectives produced by a certain manufacturing process is 0.08 and a sample of 400 is chosen, what is the probability that the proportion of defectives in the sample is greater than 0.10? Since n  p = 400(0.08) = 32 > 10 and n(1-p) = 400(0.92) = 368 > 10, it’s reasonable to use the normal approximation.

38. Example (Continued)

39. Example Suppose 3% of the people contacted by phone are receptive to a certain sales pitch and buy your product. If your sales staff contacts 2000 people, what is the probability that more than 5% of the people contacted will purchase your product? Clearly p = 0.03 and p = 100/2000 = 0.05 so

40. Example - Continued If your sales staff contacts 2000 people, what is the probability that less than 2.5% of the people contacted will purchase your product? Now p = 0.03 and p = 50/2000 = 0.025 so

41. Review Central limit theorem for the sample means If the sample size is sufficiently large, then the sampling distribution of the sample means is approximately normal regardless of the shape of the the parent distribution. Rule of thumb: it is generally safe to apply the CLT for means when n  30. “Central limit theorem” for the sample proportions The sampling distribution of sample proportions is approximately normal when np  10 and n(1-p)  10