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D ID YOU PREDICT THE D UKE U PSET IN THE FIRST ROUND OF THE NCAA TOURNAMENT ? Slide Yes 2. No 3. Who is Duke? 4. What is NCAA tournament?

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U PCOMING WORK Quiz #4 in class this Thursday HW #8 due next Monday Part 3 of Data Project due April 3rd

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P OSSIBLE T ESTS One-proportion z-test Two-proportion z-test One-sample t-test for mean Two-sample t-test for differences of means Slide 1- 3

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E XAMPLES OF O NE - 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- 4

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E XAMPLES OF T WO - 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- 5

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E XAMPLES OF O NE -S AMPLE 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- 6

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E XAMPLES 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- 7

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

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T WO - 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- 9

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M ODELING THE D ISTRIBUTION OF S AMPLE P ROPORTIONS What proportion of the population thinks Kentucky 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? Slide 1- 10

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S AMPLING D ISTRIBUTION OF P ROPORTIONS - W HAT 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. Slide 1- 11

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A N OTE ON THE S TANDARD D EVIATION 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

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M ODELING THE D ISTRIBUTION OF S AMPLE P ROPORTIONS ( CONT.) A picture of what we just discussed is as follows: Slide 1- 13

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T HE S AMPLING D ISTRIBUTION M ODEL FOR A P ROPORTION ( 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:. Slide 1- 14

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S AMPLING D ISTRIBUTION FOR A P ROPORTION - P ROBLEM 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 sampled proportion of students, who believe Duke will win, is greater than 90%? Slide 1- 15

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W HAT IS THE MEAN AND STANDARD DEVIATION OF THE PROPORTION OF THE POPULATION THAT THINKS KY WILL WIN ? 1. 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 Slide 1- 16

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W HAT ’ S THE Z- SCORE FOR A PROPORTION OF 0.90? 1. Z=( )/ Z=( )/ Z=( )/ Z=( )/0.6 Slide 1- 17

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W HAT ’ S THE PROBABILITY THAT THE SAMPLED PROPORTION OF STUDENTS WHO BELIEVE KY WILL WIN IS GREATER THAN 90%? Slide 1- 18

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M ODELING THE D ISTRIBUTION OF S AMPLE P ROPORTIONS ( CONT.) A picture of what we just discussed is as follows: Slide 1- 19

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S AMPLING D ISTRIBUTIONS - P ROPORTION VS. M EAN 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. Slide 1- 20

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B UT W HICH N ORMAL ? ( CONT.) Slide The Normal model for the sampling distribution of the mean has a standard deviation equal to where σ is the population standard deviation.

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S AMPLING D ISTRIBUTION FOR A M EAN - P ROBLEM 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- 22

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W HAT IS THE Z- SCORE FOR $40,000? 1. 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) Slide 1- 23

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W HAT ’ S THE PROBABILITY THAT THE SAMPLE ’ S AVERAGE SALARY IS LESS THAN $40,000? Slide 1- 24

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T HE F UNDAMENTAL T HEOREM OF S TATISTICS 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. Slide 1- 25

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W HEN TO A PPLY THE N ORMAL M ODEL ? Random Observations Independent Trials Each sample is the same size (n) Sample size is appropriately large Slide 1- 26

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A SSUMPTIONS AND C ONDITIONS ( 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. Slide 1- 27

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A SSUMPTIONS AND C ONDITIONS ( 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. Slide 1- 28

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KY P ROBLEM 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- 29

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A RE THE CONDITIONS NECESSARY TO USE THE NORMAL MODEL MET ? 1. 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 Slide 1- 30

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HW - P ROBLEM 1 State police believe that 40% of the drivers traveling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 20 cars. Using the rule, draw and label the distribution of the proportion of these cars the police will observe speeding. Slide 1- 31

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D O YOU THINK THE APPROPRIATE CONDITIONS NECESSARY FOR YOU ANALYSIS ARE MET ? 1. 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. Slide 1- 32

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T HE R EAL W ORLD AND THE M ODEL W ORLD 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.

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S TANDARD D EVIATION Slide Both of the sampling distributions we’ve looked at are Normal. For proportions For means

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E XAMPLE – C OIN T OSS You flip a coin 25 times and get a head 18 times (72% of the time). Is the coin fair? p=.5 SD=.1 What if you flipped it 64 times? SD=.0625 Slide 1- 35

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S TANDARD D EVIATION VS. S TANDARD E RROR 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. Slide 1- 36

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S TANDARD E RROR Slide For a sample proportion, the standard error is For the sample mean, the standard error is

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HW - P ROBLEM 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- 38

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C AN THIS COLLEGE BRAG ABOUT ITS RETENTION RATE ? 1. 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. Slide 1- 39

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HW - P ROBLEM 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- 40

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W HAT IS THE PROBABILITY THAT THE NEWSPAPER ’ S SAMPLE WILL LEAD IT TO PREDICT DEFEAT ? Slide 1- 41

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HW - P ROBLEM 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- 42

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W HAT IS THE PROBABILITY THAT THE SHIPMENT WILL BE ACCEPTED ANYWAY ? Slide 1- 43

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S TEPS TO CALCULATE THE PROBABILITY FROM A SAMPLING DISTRIBUTION Calculate mean or proportion from the sample. Determine if you know the true proportion or mean. 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- 44

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S AMPLING 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.

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U PCOMING WORK Quiz #4 in class this Thursday HW #8 due next Monday Part 3 of Data Project due April 3rd

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