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Provide Answers for each required question below:

Chapter 14 Problems: 1, 3, 28

Chapter 15 Problems: 1, 14

Chapter 17 Problems: 7, 8, 9, 11, 15, 40, 45

14. 1 Suppose that you measure the heights of an SRS of 400 eight-year-old American girls, a population with mean μ = 140 cm and standard deviation σ = 8 cm. The mean x¯ of the 400 heights will vary if you take repeated samples.

a) The sampling distribution of x¯ is approximately Normal. It has mean μ = 140 cm. What is its standard deviation?

b) Sketch the Normal curve that describes how x¯ varies in many samples from this population. Mark the mean μ = 140. According to the 68–95–99.7 rule, approximately 95% of all the values of x¯ fall within ___________ cm of the mean. What is the missing number? Call it m for “margin of error.” Shade the region from the mean minus m to the mean plus m on the axis of your sketch, as in Figure 14.1.

c) Whenever x¯ falls in the region you shaded, the true value of the population mean, μ = 140, lies in the interval between x¯ – m and x¯ + m. Draw that interval below your sketch for one value of x¯ inside the shaded region and one value of x¯ outside the shaded region.

d) In what percent of all samples will the confidence interval x¯ ± m capture the true mean μ = 140?

14.3 A 2012 Gallup survey of a random sample of 1014 American adults indicates that American families spend, on average, $151 per week on food. The report further states that, with 95% confidence, this estimate has a margin of error of ±$7.

a) This confidence interval is expressed in the following form: “estimate ± margin of error” What is the range of values (lower bound, upper bound) that corresponds to this confidence interval?

b) What is the parameter captured by this confidence interval? What does it mean to say that we have “95% confidence” in this interval?

14.28 You calculate a 95% confidence interval of 27±2 centimeters (cm) for the mean needle length of Torrey pine trees. You ask friends to explain this result.

a) One friend believes it means that “95% of all Torrey pine needles have lengths between 25 and 29 cm.” Is that right? Explain your answer.

b) Another friend thinks it means that “we can be 95% confident that the true mean needle length of Torrey pine trees is 27 cm.” Is that right? Explain your answer.

15.1 The New England Journal of Medicine posts its peer-reviewed articles and editorials on its website. An opt-in poll was featured next to an editorial about the regulation of sugar-sweetened beverages. The poll asked, “Do you support government regulation of sugar-sweetened beverages?” Readers just needed to click on a response (yes or no) to become part of the sample. The poll stayed open for several weeks in October 2012. Of the 1290 votes cast, 864 were “yes” responses.

a) Would it be reasonable to calculate from these data a confidence interval for the percent answering “yes” in the American population? Explain your answer.

b) Would it be reasonable to calculate from these data a confidence interval for the percent answering “yes” among online readers of the New England Journal of Medicine? Explain your answer.

15.14 IQ scores are typically Normally distributed within a homogeneous population. How large a sample of schoolgirls in Example 14.3 (page 354) would be needed to estimate the mean IQ score μ within ± 5 points with 99% confidence, assuming that σ = 15 in this population? (Note that the z multiplier for a 99% confidence level is z* = 2.576)

17.7 Researchers studied the morphological attributes of monarch butterflies (Danaus plexippus), a species that undertakes large seasonal migrations over North America. They measured the forewing weight (in milligrams, mg) of a sample of 92 monarch butterflies, all of which had been reared in captivity in identical conditions. Figure 17.4 shows the output from the statistical software JMP. (The data are also available in the Large.Butterfly data file if you wish to practice working with your own software.) Estimate with 95% confidence the mean forewing weight of monarch butterflies reared in captivity.

17.8 A student in wildlife management studied trout habitats in the upper Shavers Fork watershed in West Virginia. The springtime water pH of 29 randomly selected tributary sample sites was found to have the following values:

6.2 6.3 5.0 5.8 4.6 4.7 4.7 5.4 6.2 6.0

5.4 5.9 6.2 6.1 6.0 6.3 6.2 5.8 6.2 6.3

6.3 6.3 6.4 6.5 6.6 6.1 6.3 4.4 6.7

Use a 90% confidence interval to estimate the mean springtime water pH of the tributary water basin around the Shavers Fork watershed.

17.9 Do the data of Exercise 17.8 give good reason to think that the springtime water in the tributary water basin around the Shavers Fork watershed is not neutral (a neutral pH is the pH of pure water, pH=7)? Conduct a hypothesis test.

17.11 A study examined the accuracy of consumer technology used for tracking physical activity. The report states that when a random sample of 28 healthy adults each walked exactly 500 steps on a treadmill, the Fitbit Flex reported an average of 466 steps with a standard deviation of 93 steps. The iPhone 5s Moves app, by comparision, reported an average of 531 steps with a standard deviation of 55 steps. Do these results provide significant evidence that these fitness tools report inaccurate numbers of steps on average? Figure 17.7 gives the Minitab output for each technology.

a. State the null and alternative hypotheses for each technology.

b. Give the test statistic and P-value for the Fitbit Flex and conclude in context.

c. Give the test statistic and P-value for the iPhone 5s Moves app and conclude in context.

17.15 Essential tremor is a neurological movement disorder characterized by involuntary rhythmic movement that typically interferes with the full use of the arms and hands. Example 7.8 (page 175) described a pilot experiment examining the effectiveness of a noninvasive handheld device using active cancellation of tremor (ACT) technology to stabilize tremor-induced motion in patients diagnosed with essential tremor. Here is the tremor amplitude measured (in centimeters) with a precision accelerometer for each of 11 subjects when performing a spoon-use task with the ACT device turned, in random order, once on and once off.

Subject 1 2 3 4 5 6 7 8 9 10 11

ACT off 1.2 1.6 2 1 1.7 1.9 0.8 1.7 1.6 0.5 1.4

ACT on 0.3 0.2 0.6 0.2 0.5 0.4 0.2 0.6 0.3 0.2 0.1

a. Figure 17.9 shows the R output for the 95% confidence interval for the mean difference in tremor amplitude based on these data. Interpret this confidence interval in context.

b. Compute the 11 paired differences in tremor amplitude (off minus on), then obtain their mean and standard deviation.

c. The critical value for a 95% confidence level under the t(10) distribution is 2.228. You may want to verify that this value is correct from either Table C in the back of the book or statistical software. Using the formula on page 422, compute the margin of error of a 95% confidence interval for the mean difference in tremor amplitude. Verify that the resulting confidence interval agrees with the R output.

17.40 Nondigestible oligosaccharides are known to stimulate calcium absorption in rats. A double-blind, randomized experiment investigated whether the consumption of oligofructose similarly stimulates calcium absorption in healthy male adolescents 14 to 16 years old. The subjects took a pill for nine days and had their calcium absorption tested on the last day. The experiment was repeated three weeks later. Some subjects received the oligofructose pill in the first round and then a pill containing sucrose (which served as a control)in the second round. The order was switched for the remaining subjects. Here are the fractional calcium absorption data (in percent of intake) for 11 subjects:

Subject 1 2 3 4 5 6

Control 78.4 76.6 57.4 51.5 49.0 46.6

Oligofructose 62.0 95.1 46.5 49.4 89.7 43.8

Subject 7 8 9 10 11

Control 44.2 42.9 37.2 34.1 24.6

Oligofructose 50.3 51.6 66.6 52.7 54.0

a. Examine the data. Is it reasonable to use the t procedures?

b. Do the data support the hypothesis that oligofructose facilitates calcium absorption?

17.45 Can bacteria evolve a preference for the pH of their environment? An evolutionary biologist examined the relative fitness of Escherichia coli bacteria grown for 2000 generations (about 300 days) at stressful acidic pH 5.5 and their parental generation grown and preserved at pH 7.2. Both types were later grown together in an acidic medium, and their relative fitness was computed. The experiment was replicated with 6 different lines of E. coli, giving the following relative fitness values:

1.24 1.22 1.23 1.24 1.18 1.09

A relative fitness of 1 indicates that both bacteria types are equally fit. A relative fitness larger than 1 indicates that the acid-evolved line is more fit than the parental line kept at neutral pH when both are grown in acidic conditions (that is, the acid-evolved bacteria grew the most). Do the data provide evidence that bacteria evolved in acidic pH are better adapted to acidic conditions?