Comparing Two Means

Who’s Taller at Ten: Boys or Girls?Based on information from the U.S. National Health and Nutrition Examination Survey (NHANES), the heights of ten-year-old girls follow a Normal distribution with mean μf = 56.4 inches and standard deviation σf = 2.7 inches. The heights of ten-year-old boys follow a Normal distribution with mean μm = 55.7 inches and standard deviation σm = 3.8 inches. A researcher takes a random sample of 12 ten-year-old girls and a separate random sample of 8 ten-year-old boys in the United States. After analyzing the data, the researcher reports that the mean height Xm of the boys is larger than the mean height Xf of the girls.(a) Describe the shape, center, and spread of the sampling distribution of Xf − Xm.(b) Find the probability of getting a difference in sample means Xf − Xm that’s less than 0. Show your work 0.3264

(c) Does the result in part (a) give us reason to doubt the researcher’s stated results? Explain.

Potato chipsA potato chip manufacturer buys potatoes from two different suppliers, Riderwood Farms and Camberley, Inc. The weights of potatoes from Riderwood Farms are approximately Normally distributed with a mean of 175 grams and a standard deviation of 25 grams. The weights of potatoes from Camberley, Inc. are approximately Normally distributed with a mean of 180 grams and a standard deviation of 30 grams. When 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 the potatoes in the sample from Riderwood Farms, was higher than the average weight of the potatoes in the sample from Camberley, Inc. .

(a) Describe the shape, center, and spread of the sampling distribution of . (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?

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Plastic Grocery BagsDo plastic bags from Target or plastic bags from Bashas hold more weight? A group of AP Statistic 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 10,896Bashas: 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 Bashas.

We are 99% confident that the interval from –101, 7285 grams captures the true difference in the mean capacity for plastic grocery bags from Target and from Bashas.

(b) Does your interval provide convincing evidence that there is a difference in the mean capacity among the two stores?(b) Since the interval includes 0, it is plausible that there is no difference in the two means. Thus, we do not have convincing evidence that there is a difference in mean capacity. However, if we increased the sample size we would likely find a convincing difference since it seems pretty clear that Target bags have a bigger capacity

The U.S. Department of Agriculture (USDA) conducted a survey to estimate the average price of wheat in July and in September of the same year. Independent random samples of wheat producers were selected for each of the two months. Here are summary statistics on the reported price of wheat from the selected producers, in dollars per bushel:

Construct and interpret a 99% confidence interval for the difference in the mean wheat price in July and in September.

Calcium and Blood Pressure

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. The relationship was strongest for black men.

Such observational studies do not establish causation. Researchers therefore designed a randomized comparative experiment.

The subjects were 21 healthy black men who volunteered to take part in the experiment. They were randomly assigned to two groups: 10 of the men received a calcium supplement for 12 weeks, while 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 (top number) blood pressure for a subject after 12 weeks, in millimeters of mercury. An increase appears as a negative response. Here are the data:

Do the data provide sufficient evidence to conclude that a calcium supplement reduces blood pressure more than a placebo?

Technology gives a P-value of 0.0644 for the test in part (a). Interpret this value in context.

The stronger picker-upper?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, 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, 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, 87

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, 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, 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.

AP EXAM TIP When a significance test leads to a fail to reject H0 decision, as in the previous example, be sure to interpret the results as “We don’t have enough evidence to conclude Ha.” Saying anything that sounds like you believe H0 is (or might be) true will lead to a loss of credit.

AP Exam Common ErrorMany students have trouble deciding when it is appropriate to use a two-sample t test and when is it appropriate to use a paired t test. The key distinction is how the data were produced. In an experiment, if groups were formed using a completely randomized design, then a two-sample t test is the right choice. However, if subjects were paired and then split at random into the two treatment groups, or if each subject received both treatments, then a paired t test is appropriate.

In each of the following settings, decide whether you should use paired t procedures or two-sample t procedures to perform inference. Explain your choice.

(a) To test the wear characteristics of two tire brands, A and B, one brand A tire is mounted on one side of each car in the rear, while a Brand B tire is mounted on the other side. Which side gets which brand is determined by flipping a coin. The same procedure is used on the front.

Paired t procedures. This is a matched pairs experiment, with the two treatments (Brand A and Brand B) being randomly assigned to the front pair and rear pair of wheels on each car

(b) Can listening to music while working increase productivity? Twenty factory workers agree to take part in a study to investigate this question. Researchers randomly assign 10 workers to do a repetitive task while listening to music and the other 10 workers to do the task in silence.

(b) Two-sample t procedures. The data are being produced using two independent groups of workers in a randomized experiment.

Exercises on page 626 & 652# 29-32, 35, 37, 57, 39, 41, 43, 45, 51, 53, 59, 65, 67-70