Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley How many hours of sleep did you get last night? Slide < >9

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Possible Tests One-proportion z-test Two-proportion z-test One-sample t-test for mean Two-sample t-test for differences of means Slide 1- 2

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples of One-proportion test Everyone (100%) believes in ghosts More than 10% of the population believes in ghosts Less than 2% of the population has been to jail 90% of the population wears contacts Slide 1- 3

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples of Two-proportion tests Women believe in ghosts more than men Blacks believe in ghost more than whites People who have been to jail believe in ghosts more than people who haven’t been to jail Women smoke more than men Women use facebook in the bathroom more than men Slide 1- 4

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples of One-Sample t-test All Priuses have fuel economy > 50 mpg Ford Focuses get 5 mpg on average The average starting salary for ISU graduates >$100,000 The average cholesterol level for a person with diabetes is 240. Slide 1- 5

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples of two-sample t-test The MPG for the Prius is greater than the MPG for the Ford Focus ISU male graduates have a greater starting salary than women The cholesterol levels are the same for people with and without diabetes. Slide 1- 6

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 18 Sampling Distribution Models

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Two-types of sampling distribution: Proportion: parameter is p Examples: p=proportion of people who have believe in ghosts p=proportion of cars made in Japan p=proportion of internet sales that are shipped on time Mean: parameter is μ Examples: μ= Average level of monoxide emitted from a car μ=Average payoff in a game of craps μ=Average starting salary for ISU graduate Slide 1- 8

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 9 Modeling the Distribution of Sample Proportions What proportion of the population thinks Duke will win the NCAA tournament? Sample ten people, randomly Sampling Distribution is the distribution you would get if you repeatedly sample the population. It’s a theoretical distribution. What does the distribution look like?

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Sampling Distribution of Proportions - What does it look like? The histogram of the sample proportions center at the true proportion, p, in the population. As far as the shape of the histogram goes, we can simulate a bunch of random samples that we didn’t really draw. Unimodal, symmetric, and centered at p.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Modeling the Distribution of Sample Proportions (cont.) A picture of what we just discussed is as follows:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A Note on the Standard Deviation The standard deviation of a sample, s, is just the square root of the variance and is measured in the same units as the original data. The standard deviation of a sampling distribution, is σ(p)= Slide 1- 12

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Sampling Distribution Model for a Proportion (cont.) Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with Mean: Standard deviation:.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sampling Distribution for a Proportion - Problem From past experience, I have found that 60% of my students believe Duke will win the NCAA tournament. I sample ten students from Spring What’s the probability that the proportion of students who believe Duke will win is greater than 90%? Slide 1- 14

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What is the mean and standard deviation of the proportion of the population that thinks Duke will win? Slide Mean = 0.6 SD= 0.6* Mean = 0.6 SD= sqrt(0.6*0.4) 3. Mean = 0.6 SD= sqrt(0.6*0.4/10) 4. Mean = 0.6 SD= 0.6*0.4/10

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What’s the Z-score for a proportion of 0.90? Slide Z=( )/ Z=( )/ Z=( )/ Z=( )/0.6

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What’s the probability that the proportion of students who believe Duke will win is greater than 90%? Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Modeling the Distribution of Sample Proportions (cont.) A picture of what we just discussed is as follows:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Sampling Distributions - Proportion vs. Mean The CLT says that the sampling distribution of any mean or proportion is approximately Normal. But which Normal model? For proportions, the sampling distribution is centered at the population proportion. For means, it’s centered at the population mean.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide But Which Normal? (cont.) The Normal model for the sampling distribution of the mean has a standard deviation equal to where σ is the population standard deviation.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sampling Distribution for a Mean - Problem From past experience, the average starting salary of an ISU graduate is $52,000 with a SD of $5,000 I survey 100 graduates. What’s the probability that their average salary is less than $40,000? Slide 1- 21

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What is the Z-score for $40,000? Slide Z=(40,000-52,000)/5,000/ Z=(40,000-52,000)/5,000/ Z=(40,000-52,000)/ sqrt(25,000/100) 4. Z=(40,000-52,000)/sqrt(5,000/100)

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What’s the probability that their average salary is less than $40,000? Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Fundamental Theorem of Statistics The Central Limit Theorem (CLT) The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide When to Apply the Normal Model? Random Observations Independent Trials Each sample is the same size (n) Sample size is appropriately large

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Assumptions and Conditions (cont.) 1.Independence 1.Randomization Condition: The sample should be a simple random sample of the population. 1.Unbiased 2.Representative of the Population 2.10% Condition: If sampling has not been made with replacement, then the sample size, n, must be no larger than 10% of the population.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Assumptions and Conditions (cont.) 1.Large Enough Sample 1.Success/Failure Condition: The sample size has to be big enough so that both and are greater than 10.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Duke Problem revisited From past experience, I have found that 60% of my students believe Duke will win the NCAA tournament. I sample ten students from Spring Can I use the Normal model to approximate a sampling distribution? Slide 1- 28

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Are the conditions necessary to use the normal model met? Slide Yes, all the conditions are met 2. No, the 10% condition is not met 3. No, the randomization condition is not met 4. No, the success/failure condition is not met 5. No, the randomization and success/failure condition are not met 6. No, none of the conditions are met

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HW - Problem 1 Slide 1- 30

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Do you think the appropriate conditions necessary for you analysis are met? Slide Yes, all the conditions are met 2. No, the 10% condition is not met 3. No, the randomization condition is not met. 4. No, the success/failure conditions are not met 5. No, none of the conditions are met.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Real World and the Model World Be careful! Now we have two distributions to deal with. The first is the real world distribution of the sample, which we might display with a histogram. The second is the math world sampling distribution of the statistic, which we model with a Normal model based on the Central Limit Theorem. Don’t confuse the two!

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Standard Deviation Both of the sampling distributions we’ve looked at are Normal. For proportions For means

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example – Coin Toss You flip a coin 25 times and get a head 70% of the time. Is the coin fair? p=.5 SD=.1 What if you flipped it 64 times? SD=.0625 Slide 1- 34

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Standard Deviation vs. Standard Error We don’t know p, μ, or σ, we’re stuck, right? Nope. We will use sample statistics to estimate these population parameters. Sample statistics are notated as: s, Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Standard Error For a sample proportion, the standard error is For the sample mean, the standard error is

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HW - Problem 4 The national freshman-to-sophomore retention rate has held steady at 74%. Acme College has 490 of the 592 freshman return as sophomores. Does this college have the right to brag that it has an unusually high retention rate? Slide 1- 37

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Can this college brag about its retention rate? Slide Yes, b/c their retention rate is not more than 3 SD above the expected rate. 2. Yes, b/c their retention rate is more than 4 SD above the expected rate. 3. No, b/c their retention rate is not more than 3 SD above the expected rate. 4. No, b/c their retention rate is more than 4 SD above the expected rate.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HW - Problem 5 Just before a referendum on a recycling mandate, a local newspaper polls 358 voters to predict whether the mandate will pass. Suppose the mandate has the support of 53% of the voters. What is the probability that the newspaper’s sample will lead it to predict defeat? Slide 1- 39

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What is the probability that the newspaper’s sample will lead it to predict defeat? Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HW - Problem 6 When a truck load of apples arrives at a packing plant, a random sample of 125 is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 5% of the sample is unsatisfactory. Suppose that in fact 9% of the apples on the truck do not meet the desired standard. What is the probability that the shipment will be accepted anyway. Slide 1- 41

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What is the probability that the shipment will be accepted anyway? Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Steps to calculate the probability from a sampling distribution Calculate mean or proportion from the sample. Calculate the standard error Determine the ‘standard’ you are comparing your sample to. Calculate z-score of that ‘standard’ Find percentile of that z-score. Slide 1- 43

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sampling distributions are awesome! Sampling distribution models are important because they act as a bridge from the real world of data to the imaginary model of the statistic and enable us to say something about the population when all we have is data from the real world. Slide 1- 44