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Section 10.1 - Comparing Two Proportions (pp. 609-629)The Sampling Distribution of a Difference between Two ProportionsThe Sampling Distribution of p1-p2Choose an SRS of size n1 from Population 1 with proportion of successes p1 and an independent SRS of size n2 from Population 2 with proportion of successes p2.Shape: Center:Spread:Example – Nathan and Kyle both work for the Department of Motor Vehicles (DMV), but in different states. In Nathan’s state, 80% of the registered cars are made by American manufacturers. In Kyle’s state, only 60% of the registered cars are made by American manufacturers. Nathan selects a random sample of 100 cars from his state and Kyle selects a random sample of 70 cars from his state. Let pN-pK be the difference in the sample proportion of cars made by American manufacturers.What is the shape of the sampling distribution of pN-pK? Why?Find the mean of the sampling distribution. Show work.Find the standard deviation of the sampling distribution. Show work.Confidence Intervals for p1-p2EstimateTwo-sample z interval for p1-p2 (2-PropZInt)An approximate level C confidence interval for p1-p2 isp1-p2=±z*p11-p1n1+p21-p2n2where z* is the standard Normal critical value. Random: The data are producd by a random sample of size n1 from population 1 and a random sample of size n2 from population 2 or by two groups of size n1 and n2 in a randomized experiment.Normal: The counts of “successes” and “failures” in each sample or group -- n1p1, n1(1-p1), n2p2, n2(1-p2) – are at least 10.Independent: Both the samples or groups themselves and the individual observations in each sample or group are independent. When sampling without replacement, check that the two populations are at least 10 times as large as the corresponding samples (the 10% condition).Example - Many news organizations conduct polls asking adults in the US if they approve of the job the President is doing. How did President Obama’s approval rating change from August 2009 to September 2010? According to a CNN poll of 1024 randomly selected US adults on September 1-2, 2010, 50% approved of Mr. Obama’s performance. A CNN poll of 1010 randomly selected US adults on August 28-30, 2009, showed that 53% approved of Mr. Obama’s performance.(1) Use the results of these polls to construct and interpret a 90% confidence interval for the change in Mr. Obama’s approval rating among all US adults.(2) Based on your interval, is there convincing evidence that Mr. Obama’s job approval rating changed between August 2009 and September 2010?Significance Tests for p1-p2TestTwo-sample z test for p1-p2 (2-PropZTest)Significance tests of H0: p1 – p2 = 0 use the pooled (combined) sample proportionpc=count of successes in both samples combinedcount of individuals in both samples combined=X1+X2n1+n2The two-sample z test for p1-p2 uses the test statisticz=p1-p2-0pc1-pcn1+pc1-pcn2with P-values calculated from the standard Normal distribution.Random: The data are producd by a random sample of size n1 from population 1 and a random sample of size n2 from population 2 or by two groups of size n1 and n2 in a randomized experiment.Normal: The counts of “successes” and “failures” in each sample or group -- n1p1, n1(1-p1), n2p2, n2(1-p2) – are at least 10.Independent: Observations and independent samples or groups; 10% condition if sampling without replacementExample - Are teenagers going deaf? (“Say what?”) In a study of 3000 randomly selected teenagers in 1988-1994, 15% showed some hearing loss. In a similar study of 1800 teenagers in 2005-2006, 19.5% showed some hearing loss. (Arizona Daily Star, August 18, 2010)Do these data give convincing evidence that the proportion of all teens with hearing loss has increased? Between the two studies, Apple introduced the iPod. If the results of the test are statistically significant, can we blame iPods for the increased hearing loss in teenagers?The Sampling Distribution of a Difference between Two MeansThe Sampling Distribution of x1-x2Choose an SRS of size n1 from Population 1 with mean 1 and standard deviation 1 and an independent SRS of size n2 from Population 2 with mean 2 and standard deviation 2.Shape:Center:Spread:Example - The Hyena Potato Chip Company buys potatoes from two different suppliers, Riderwood Farms and Camberley, Inc. The weights of the potatoes from Riderwood are approximately Normally distributed with a mean of 175 grams and a standard deviation of 25 grams. The weights of the potatoes from Camberley are approximately Normally distributed with a mean of 180 grams and a standard deviation of 30 grams. When the shipments arrive at the factory, inspectors randomly select a sample of 20 potatoes from each shipment and weigh them. They are surprised when the average weight of potatoes from Riderwood xr is higher than the average weight of the potatoes from Camberley xc .a. Describe the shape, center and spread of the sampling distribution of xc-xr .b. Find the probability that the mean weight of the Riderwood sample is larger than the mean weight of the Camberley sample. Should the inspectors have been surprised?Confidence Intervals for x1-x2EstimateTwo –sample t interval for 1-2 (2-SampTInt)x1-x2±t*s12n1+s22n1df = min(n1 - 1, n2 - 1)Random: Data from random samples or randomized experimentNormal: Population distributions Normal or large samples (n1≥30, n2≥30)Independent: Observations and independent samples or groups; 10% condition if sampling without replacementExample - Do plastic bags from Target or plastic bags from Walmart hold more weight? A group of AP Statistics students decided to investigate by filling a random sample of 5 bags from each store with common grocery items until the bags ripped. Then they weighed the contents of items in each bag to determine its capacity. Here are the results in grams:Target1257213999112151544710896Walmart955210896698387679972a. Construct and interpret a 99% confidence interval for the difference in the mean capacity of plastic grocery bags from Target and Walmart.Significance Tests for x1-x2TestTwo-sample t test for 1-2 (2-SampTTest)t=(x1-x2)-(μ1-μ2)s12n1+s22n2df = min(n1 - 1, n2 - 1)Random: Data from random samples or randomized experimentNormal: Population distributions Normal or large samples (n1≥30, n2≥30)Independent: Observations and independent samples or groups; 10% condition if sampling without replacementExample - In commercials for Bounty paper towels, the manufacturer claims that they are the “quicker picker-upper.” But are they also the stronger picker-upper? Two AP Statistics students selected a random sample of 30 Bounty paper towels and 30 generic paper towels and measured their strength when wet. To do this, they uniformly soaked each paper towel with 4 ounces of water, held two opposite edges of the paper towel, and counted how many quarters each paper towel could hold until ripping, alternating brands. Here are the results:Bounty: 106 111 106 120 103 112 115 125 116 120 126 125 116 117 114 118 126 120 126 125 116 117 114 118 126 120 115 116 121 113 111 128 124 125 127 123 115 114Generic:77 103 89 79 88 86 100 90 81 84 84 96 87 79 90 86 88 81 91 94 90 89 85 83 89 84 90 100 94 87a. Use a significance test to determine whether there is convincing evidence that wet Bounty paper towels can hold more weight, on average, than wet generic paper towels can.b. Interpret the P-value from part b in the context of the question.c. Does the interval provide convincing evidence that there is a difference in the mean capacity between the stores? ................
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