Section 13.1: Comparing Two Means Characteristics of two sample problems: AP Stats Chapter 13 What are the three conditions for comparing two means? When two population means are compared you can do one of two things: What is the test statistic for comparing two means? How do I do it on the calculator?
Example 1: The stronger picker-upper? In commercials for Bounty paper towels, the manufacturer claims that they are the quicker pickerupper. But are they also the stronger picker upper? Two AP Statistics students, Wesley and Maverick, decided to find out. They selected a random sample of 30 Bounty paper towels and a random sample of 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 their results: Bounty: 106, 111, 106, 120, 103, 112, 115, 125, 116, 120, 126, 125, 116, 117, 114 118, 126, 120, 115, 116, 121, 113, 111, 128, 124, 125, 127, 123, 115, 114 Generic: 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, 87 (a) Display these distributions using parallel boxplots and briefly compare these distributions. Based only on the boxplots, discuss whether or not you think the mean for Bounty is significantly higher than the mean for generic. (b) Use a significance test to determine if there is convincing evidence that wet Bounty paper towels can hold more weight, on average, than wet generic paper towels. (c) Interpret the P-value from (b) in the context of this question.
Example 2: Does increasing the amount of calcium in our diet reduce blood pressure? Examination of a large sample of people revealed a relationship between calcium intake and blood pressure. Such observational studies do not establish causation. Researchers therefore designed a randomized comparative experiment. The subjects were 21 men. A randomly chosen group of 10 men received a calcium supplement for 12 weeks. The control group of 11 men received a placebo pill that looked identical. The experiment was double-blind. The response variable is the decrease in systolic blood pressure for a subject after 12 weeks. An increase appears as a negative response. Take group 1 to be the calcium group and group 2 to be the placebo group. Here are the data for the 10 men in group 1 (calcium): 7-4 18 17-3 -5 1 10 11-2 and for the 11 men in group 2 (control): -1 12-1 -3 3-5 5 2-11 -1-3 Are these data good evidence that calcium decreases blood pressure in the population of men more than a placebo does?
Example 3: Poisoning by the pesticide DDT causes convulsions in humans and other mammals. Researchers seek to understand how the convulsions are caused. In a randomized comparative experiment, they compared 6 white rats poisoned with DDT with a control group of 6 unpoisoned rats. Electrical measurements of nerve activity are the main clue to the nature of DDT poisoning. When a nerve is stimulated, its electrical response shows a sharp spike followed by a much smaller second spike. Researchers wanted to know if the second spike is different in rats poisoned by DDT. The researchers measured the height of the second spike as a percent of the first spike when a nerve on the rat s leg was stimulated. For the poisoned rats the results were 12.207 16.869 25.050 22.429 8.456 20.589 The control group data were 11.074 9.686 12.064 9.351 8.182 6.642 What is the formula for a confidence interval when comparing two means? How do I do this on the calculator?
Example 4: Plastic grocery bags 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 their results, in grams: Target: 12,572 13,999 11,215 15,447 10,896 Bashas: 9552 10,896 6983 8767 9972 (a) Construct and interpret a 99% confidence interval for the difference in mean capacity of plastic grocery bags from Target and Walmart. (b) Does your interval provide convincing evidence that there is a difference in the mean capacity between the two stores?
Example 5: A man who moves to a new city sees that there are two routes he could take to work. A neighbor who has lived there a long time tells him route A will average 5 minutes faster than route B. The man decides to experiment. Each day he flips a coin to determine which way to go, driving each route 20 days. He finds that route A takes an average of 40 minutes, with standard deviation 3 minutes. Route B takes an average of 43 minutes, with a standard deviation of 2 minutes. Boxplots of the travel times are roughly symmetric and show no outliers. a) Find a 95% confidence interval for the difference in average commuting time for the two routes. b) Should the man believe the neighbor s claim that he can save an average of 5 minutes a day by always driving route A? Explain. Section 13.2: Comparing Two Proportions What is the formula for a two-sample z interval for p1 p2? Is this on the formula sheet? What are the conditions for calculating a two-sample z interval for p1 p2? How are these different than the conditions for a one-sample z interval for p?
How do I do this on the calculator? Example 6: Presidential approval Many news organizations conduct polls asking adults in the United States 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 U.S. adults on September 1 2, 2010, 50% approved of Obama s job performance. A CNN poll of 1010 randomly selected U.S. adults on August 28 30, 2009, showed that 53% approved of Obama s job performance. (a) Explain why we should use a confidence interval to estimate the change in Obama s approval rating rather than just saying that his approval rating went down 3 percentage points. (b) Use the results of these polls to construct and interpret a 90% confidence interval for the change in Obama s approval rating among all U.S. adults. (c) Based on your interval, is there convincing evidence that Obama s job approval rating has changed? Example 7: In October 2000 the U.S. Dept. of Commerce reported the results of a large scale survey on high school graduation. Researchers contacted more than 25,000 Americans aged 24 years to see if they had finished high school. 84.9% of the 12, 460 males and 88.1% of the 12, 678 females indicated that they had high school diplomas. a) Create a 95% confidence interval for the difference in graduation rates for males and females. Does this provide strong evidence that girls are more likely than boys to complete high school?
What is the test statistic for a two-sample z test for a difference in proportions? Is this on the formula sheet? How do I do this on the calculator? Example 8: Hearing loss Are teenagers going deaf? 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. (These data are reported in Arizona Daily Star, August 18, 2010.) (a) Do these data give convincing evidence that the proportion of all teens with hearing loss has increased? (b) 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?
Example 9: To study the long term effects of preschool programs for poor children, the High/Scope Educational Research Foundation has followed two groups of Michigan children since early childhood. A control group of 61 children represents Population 1, poor children with no preschool. Another group of 62 from the same area and similar backgrounds attended preschool as 3 and 4 year olds. This is a sample from Population 2, poor children who attend preschool. One response variable of interest is the need for social services as adults. In the past ten years, 38 of the preschool sample and 49 of the control sample have needed social services (mainly welfare). Does this study provide significant evidence that preschool reduces the later need for social services? Example 10: The movie A Civil Action tells the story of a major legal battle that took place in the small town of Woburn, Massachusetts. A town well that supplied water to East Woburn residents was contaminated by industrial chemicals. During the period that residents drank water from this well, a sample of 414 births showed 16 birth defects. On the west side of Woburn, a sample of 228 babies born during the same time period showed 3 with birth defects. The plaintiffs suing the companies responsible for the contamination claimed that these data show that the rate of birth defects was significantly higher in East Woburn than in West Woburn. How strong is the evidence supporting this claim? What should the judge in this case conclude?