# MATHEMATICS

HAPTER 10:

15. The highest record temperature during the month of July for a given year in Death Valley, in California, has an approximately normal distribution with a mean of 123.8oF(!) and standard deviation of 3.1oF.

a.       What is the probability for a given year that the temperature never exceeds 120oF in a given July in Death Valley?

b. What is the probability that the temperature in Death Valley goes about 128oF during July in a randomly chosen year?

26. The crab spider, Thomisus, spectabilis, sits on flowers and preys upon visiting honeybees, as shown in the photo at the beginning of the chapters. (Remember this next you sniff a wildflower.) Do honeybees distinguish between flowers that have crab spiders and flowers that do not? To test this, Heiling et al. (2003) gave 33 bees a choice between two flowers: one had a crab spider and the other did not. In 24 of the 33 trials, the bees picked the flower that has the spider. In the remaining nine trials, the bees chose the spider-less flower. With these data, carry out the appropriate hypothesis test, using an appropriate approximation to calculate P.

27. The following table lists the mean and standard deviation of several different normal distributions. in each case, a sample of 20 individuals was taken, as well as a sample of 50 individuals. For each sample, calculate the probability that the mean of the sample was less than the given value.

CHAPTER 11:

14. Astronauts increased in height by an average of approximately 40 mm (about an inch and a half) during the Apollo-Soyuz missions, due to the absence of gravity compressing their spines during their tie in space. Does something similar happen here on Earth? An experiment supported by NASA measured the heights of six men immediately before going to bed, and then again after three days of bed rest. One average, they increased in height by 14mm, with standard deviation of 0.66mm. Find the 95% confidence interval for the change in height after three days of best rest.

16. In the Northern Hemisphere, dolphins swim predominately in a counterclockwise direction while sleeping. A group of research wanted to know whether the same was true for dolphins in the Southern Hemisphere . They watched eight sleeping dolphins and recorded the percentage of time that the dolphins swam clockwise. Assume that this is a random sample and that this variable has a normal distribution in the population. These data are as follows:

77.7, 84.8, 79.4, 84.0, 99.6, 93.6, 89.4, 97.2

a. What is the mean percentage of clockwise swimming for Southern Hemisphere dolphins?

b. What is the 95% confidence interval for the mean time swimming clockwise in the south Hemisphere dolphins?

c. What is the 99% confidence interval for the mean time swimming clockwise in the Southern Hemisphere dolphins?

d. What is the best estimate of the standard deviation of the percentage of clockwise swimming?

e. What is the median of the percentage of clockwise swimming?

21. Hurricanes Katrina and Rita caused the flooding of large parts of New Orleans, leaving behind large amounts of new sediment. Before the hurricanes, the soils in New Orleans were known to have high concentrations of lead, which is a dangerous environmental toxin. Forty six sites has been monitored before the hurricanes for soil lead content, as measured in mg/kg. And the soil from each of these sites was measured again after the hurricanes. The data given below show the log of the ratio of the soil lead content after the hurricanes and the soil lead content before the hurricanes –we’ll call this variable the “change in soil lead.” (Therefore, numbers less than zero show a reduction in soil lead content after the hurricanes. and numbers greater than zero show increases.) This log ratio has an approximately normal distribution.

-0.83, -0.18, 0.14, -1.46, -0.48, -1.04, 0.25, -0.34, -0.81, -0.83, -0.60, 0.34, -0.75, 0.37, 0.26, 0.46, -0.03, -0.32, -0.53, -1.55, -0.90, -0.95, -0.13, -0.75, 0.59, -0.06, 0.39, -0.40, -0.84, -0.56, 0.44, 0.18, 0.28, -0.41, -0.26, 0.64, -0.51, -0.36, 0.49, 0.21, 0.17, 0.13, -0.63, -1.24, 0.57, -0.78.

a. Draw a graph of the data, following recommended principles of good graph design. What trend is suggested?

b. Determine the most-plausible range of values for the mean change in soil lead. Describe in words what the nature of that change is. Is an increase in soil lead consistent with the data? Is a decrease in soil lead consistent? (you must calculate the confidence interval to answer the questions).

c. Test whether mean soil lead changed after the hurricanes.

24. Without external cues such as the sun, people attempting to walk in a straight line tend to walk in circles (the accompanying image shows he paths of two participants, PS and KS, attempting to walk in a straight line in an unfamiliar forest on a cloudy day). One idea is that most individuals have a tendency to turn in one direction because internal physiological asymmetries, or because of differences between legs in length or strength. Souman et al. tested for a directional tendency by blindfolding 15 participants in a large field and asking them to walk in a straight line. The numbers below are the median change in direction (turning angle) of each of the 15 participants measured in degrees per second. A negative angle refers to a left turn, whereas a positive number indicates a right turn.

-5.19, -1.20, -0.50, -0.33, -0.15, -0.15, -0.15, -0.07, 0.02, 0.02, 0.28, 0.37, 0.45, 1.76, 2.80.

a. Draw a graph showing the frequency distribution of the data. Is a trend in the mean angle suggested?

b. Do people tend to turn in one direction (e.g. left) more on average than the other direction (e.g right)? Test whether the mean angle differs from zero.

c. Based on your results in part (b), is the following statement justified? “People do not have a tendency to turn more in one direction, on average, than the other direction.” Explain.

25. Functionally important traits in animals tend to vary little from one individual to the next within populations, possible because individuals that deviate too much from the mean die sooner or leave fewer offspring in the long run. If so, does variance in a trait rise after it becomes less functionally important? Billet et al. (2012) investigated this question with the semicircular canals (SC) of the inner ear of the three-toed sloth. Sloths move very slowly and infrequently, and the authors suggested that this behavior reduces the functional demands on this SC, whish usually provide information on angular head movement to the brain. Indeed, the motion signal from the SC to the brain may be very weak in sloths as compared to faster moving animals. The following numbers are measurements of the ratio of the length to the width of the anterior semicircular canals in seven sloths. Assume that this represents a random sample.

1.53, 1.06, 0.93, 1.38, 1.47, 1.20, 1.16

a. In related, faster-moving animals, the standard deviation of the ratio of the length to the width of the anterior semicircular canals is known to be 0.09. What is the estimate of standard deviation of this measurement in three-toed sloths?

b. Based on these data, what is the most plausible range of values for the population standard deviation in the three-toed sloth? Does this range include the known value of the standard deviation in related, faster moving species?

c. What additional assumption is required for your answer in (b)? What do you know about how sensitive the confidence interval calculation is when the assumption is not met?

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