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| m102725 | The physical fitness score for a population of police officers at a local police station is 72, with a standard deviation of 7 on a 100-point physical endurance scale. Suppose the police chief selects a sample of 49 local police officers from this population and records a mean physical fitness rating on this scale equal to 74. He conducts a one-sample z test to determine whether physical endurance increased at a .05 level of significance.
(a) State the value of the test statistic and whether to retain or reject the null hypothesis.
(b) Compute effect size using Cohen s d. |
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| m102726 | The Poisson distribution (Section 5.4) gives the probability for the number of occurrences for a "rare" event. Now, let x be a random variable that represents the waiting time between rare events. Using some mathematics, it can be shown that x has an exponential distribution. Let x > 0 be a random variable and let β > 0 be a constant. Then y = 1/ βe-x/β is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities.
If a and b are any numbers such that 0 6 a 6 b, then using some extra mathematics, it can be shown that the area under the curve above the interval [a, b] is
P(a < x < b) = e - a/β - e -b/β
Notice that by de finition, x cannot be negative, so, P(x < 0) = 0. The random variable x is called an exponential random variable. Using some more mathematics, it can be shown that the mean and standard deviation of x are
( = β and ( = β
The number e = 2.71828 . . . is used throughout probability, statistics and mathematics. The key ex is conveniently located on most calculators.
The Poisson and exponential distributions have a special relationship. Specifi cally, it can be shown that the waiting time between successive Poisson arrivals (i.e., successes or rare events) has an exponential distribution with β = 1/(, where l is the average number of Poisson successes (rare events) per unit of time. For more on this topic, please see Problem 20.
Fatal accidents on scheduled domestic passenger flights are rare events. I |
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| m102727 | The probability is 0.314 that the gestation period of a woman will exceed 9 months. In 4000 human gestation periods, roughly how many will exceed 9 months? |
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| m102728 | The probability is 0.667 that the favorite in a horse race will finish in the money (first, second, or third place). In 500 horse races, roughly how many times will the favorite finish in the money? |
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| m102729 | The probability that a single radar station will detect an enemy plane is 0.65.
(a) Quota Problem How many such stations are required for 98% certainty that an enemy plane flying over will be detected by at least one station?
(b) If four stations are in use, what is the expected number of stations that will detect an enemy plane? |
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| m102730 | The Professional Golfer s Association of America (PGA) organizes golf tournaments for professional golfers. The following table lists the longest drives, in yards, recorded during a PGA tournament for a random sample of 26 golfers. Use the technology of your choice to decide whether applying the z-interval procedure to these data is reasonable. Explain your answer. |
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| m102731 | The quantity of dissolved oxygen is a measure of water pollution in lakes, rivers, and streams. Water samples were taken at four different locations in a river in an effort to determine if water pollution varied from location to location. Location I was 500 meters above an industrial plant water discharge point and near the shore. Location II was 200 meters above the discharge point and in midstream. Location III was 50 meters downstream from the discharge point and near the shore. Location IV was 200 meters downstream from the discharge point and in midstream. The following table shows the results. Lower dissolved oxygen readings mean more pollution. Because of the dif ficulty in getting midstream samples, ecology students collecting the data had fewer of these samples. Use an a = 0.05 level of signifi cance. Do we reject or not reject the claim that the quantity of dissolved oxygen does not vary from one location to another?
In the problem, please provide the following information.
(a) What is the level of signifi cance? State the null and alternate hypotheses.
(b) Find SSTOT, SSBET, and SSW and check that SSTOT - SSBET + SSW. Find d.f.BET, d.f.W, MSBET, and MSW. Find the value of the sample test statistic F (F ratio).
What are the degrees of freedom?
(c) Find (or estimate) the P-value of the sample test statistic.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?
(e) Interpret your conclusion in the context |
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| m102732 | The random variable X is the crew size of a randomly selected shuttle mission between April 12, 1981 and July 8, 2011. Its probability distribution is as follows.
We have provided the probability distributions of the random variables considered in Exercises 5.11-5.15 of Section 5.1. For each exercise, do the following tasks.
a. Find and interpret the mean of the random variable.
b. Obtain the standard deviation of the random variable by using one of the formulas given in Definition 5.5.
c. Construct a probability histogram for the random variable; locate the mean; and show one-, two-, and three-standard-deviation intervals. |
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| m102733 | The random variable Y is the number of major hurricanes for a randomly selected year between 1851 and 2012. Its probability distribution is as follows.
We have provided the probability distributions of the random variables considered in Exercises 5.11-5.15 of Section 5.1. For each exercise, do the following tasks.
a. Find and interpret the mean of the random variable.
b. Obtain the standard deviation of the random variable by using one of the formulas given in Definition 5.5.
c. Construct a probability histogram for the random variable; locate the mean; and show one-, two-, and three-standard-deviation intervals. |
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| m102734 | The Recording Industry Association of America (RIAA) reports annual statistics on shipments and revenues. For the year 2012, RIAA reported the following data on physical shipments.
Identify the type of data provided by the information in each of the following columns of the table:
a. First
b. Second
c. Third |
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| m102735 | The report "Addiction Medicine: Closing the Gap between Science and Practice" from the National Center on Addiction and Substance Abuse at Columbia University reveals that addiction treatment is neglected by the medical system in the United States. It shows, in particular, that 16% of Americans have the disease of addiction and that 90% of them receive no form of treatment. Determine the probability that a randomly selected American has the disease of addiction and receives no form of treatment. Interpret your answer in terms of percentages. |
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| m102736 | The reproduction capability of spider mites infected with a bacteria called Cardinium was studied in the article "Cardinium Symbionts Cause Cytoplasmic Incompatibility in Spider Mites" (Heredity, Vol. 98, No. 1, pp. 13-20) by T. Gotoh et al. Infected mites were crossed with mites treated with antibiotics. The following table shows the number of eggs per female for each combination tested. Here we use "A" for antibiotic-treated, "I" for infected, "M" for male, and "F" for female.
At the 10% significance level, do the data provide sufficient evidence to conclude that a difference exists in mean number of eggs among the four combinations?
Perform a Kruskal-Wallis test by using either the critical-value approach or the P-value approach |
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| m102737 | The respiratory disease emphysema, which is most commonly caused by smoking, causes damage to the air sacs in the lungs. According to the National Center for Health Statistics report Data from the National Health Interview Survey, 1.5% of the adult American population suffer from emphysema. Of 100 randomly selected adult Americans, let X denote the number who have emphysema.
a. What are the parameters for the appropriate binomial distribution?
b. What is the parameter for the approximating Poisson distribution?
c. Compute the individual probabilities for the binomial distribution in part (a). Obtain the probabilities until they are zero to four decimal places.
d. Compute the individual probabilities for the Poisson distribution in part (b). Obtain the probabilities until they are zero to four decimal places.
e. Compare the probabilities that you obtained in parts (c) and (d). |
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| m102738 | The Reverend Thomas Bayes (1702-1761) was an English mathematician who discovered an important rule of probability (see Bayes s theorem, Appendix I, part I). A key feature of Bayes s theorem is the formula P (B) = P (B | A) ( P (A) + P (B | Ac) ( P (Ac) Explain why this formula is valid? |
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| m102739 | The rule of thumb for using the normal approximation to the binomial is that both np and n(1 − p) are 5 or greater. Why is this restriction necessary? |
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| m102740 | The same experimental design discussed in Problem 20 was used to test rats trained to climb a sequence of short ladders. Times in seconds for eight rats to perform this task are shown in the following table.
Do these data indicate that rats receiving larger rewards tend to climb the ladder in less time? Use a 5% level of signifi cance.
(a) What is the level of signifi cance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?
(b) What sampling distribution will you use? What assumptions are you making? Compute the value of the sample test statistic and corresponding t value.
(c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically signifi cant at level a?
(e) Interpret your conclusion in the context of the application. |
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| m102741 | The sample mean is an unbiased estimator of the population mean. Explain this statement. |
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| m102742 | The sample variance is an unbiased estimator of the population variance when we divide SS by df. What does this mean? |
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| m102743 | The Scottish Executive, Analytical Services Division Transport Statistics, compiles data on motorcycle casualties. During one year, monthly casualties resulting from motorcycle accidents in Scotland for built-up roads and non-built-up roads were as follows.
a. Without doing any calculations, make an educated guess at which of the two data sets, built-up or non-built-up, has the greater variation.
b. Find the range and sample standard deviation of each of the two data sets. Compare your results here to the educated guess that you made in part (a). |
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| m102744 | The second-leading genetic cause of mental retardation is Fragile X Syndrome, named for the fragile appearance of the tip of the X chromosome in affected individuals. One in 1500 males is affected worldwide, with no ethnic bias.
a. In a sample of 10,000 males, how many would you expect to have Fragile X Syndrome?
b. For a sample of 10,000 males, use the Poisson approximation to the binomial distribution to determine the probability that more than 7 of the males have Fragile X Syndrome; that at most 10 of the males have Fragile X Syndrome. |
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