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The actual tracking weight of a stereo cartridge that is set to track at \({\rm{3 g}}\) on a particular changer can be regarded as a continuous rv \({\rm{X}}\) with pdf

\({\rm{f(x) = \{ }}\begin{array}{*{20}{c}}{{\rm{k(1 - (x - 3}}{{\rm{)}}^2})}&{{\rm{2}} \le {\rm{x}} \le {\rm{4}}}\\{\rm{0}}&{{\rm{otherwise}}}\end{array}\)

a. Sketch the graph of \({\rm{f(x)}}\).

b. Find the value of \({\rm{k}}\).

c. What is the probability that the actual tracking weight is greater than the prescribed weight?

d. What is the probability that the actual weight is within \({\rm{.25 g}}\) of the prescribed weight?

e. What is the probability that the actual weight differs from the prescribed weight by more than \({\rm{.5 g}}\)?

The California State University (CSU) system consists of 23 campuses, from San Diego State in the south to Humboldt State near the Oregon border. A CSU administrator wishes to make an inference about the average distance between the hometowns of students and their campuses. Describe and discuss several different sampling methods that might be employed. Would this be an enumerative or an analytic study? Explain yourreasoning.

Once an individual has been infected with a certain disease, let \({\rm{X}}\) represent the time (days) that elapses before the individual becomes infectious. The article proposes a Weibull distribution with \({\rm{\alpha = 2}}{\rm{.2}}\), \({\rm{\beta = 1}}{\rm{.1}}\), and \({\rm{\gamma = 0}}{\rm{.5}}\).

a. Calculate \({\rm{P(1 < X < 2)}}\).

b. Calculate \({\rm{P(X > 1}}{\rm{.5)}}\).

c. What is the \({\rm{90th}}\) percentile of the distribution?

d. What are the mean and standard deviation of \({\rm{X}}\)?

The three measures of center introduced in this chapter are the mean, median, and trimmed mean. Two additional measures of center that are occasionally used are the midrange,which is the average of the smallest and largest observations, and the midfourth,which is the average of the two fourths. Which of these five measures of center are resistant to the effects of outliers and which are not? Explain your reasoning.

A certain city divides naturally into ten district neighborhoods.How might a real estate appraiser select a sample of single-family homes that could be used as a basis for developing an equation to predict appraised value from characteristics such as age, size, number of bathrooms, distance to the nearest school, and so on? Is the study enumerative or analytic?

If the amount of soft drink that I consume on any given day is independent of consumption on any other day and is normally distributed with\(\mu = 13\)oz and\(\sigma = 2\)and if I currently have two six-packs of\({\bf{16}}\)-oz bottles, what is the probability that I still have some soft drink left at the end of\({\bf{2}}\)weeks?

The article 鈥淩eliability-Based Service-Life Assessment of Aging Concrete Structures鈥� (J. Structural Engr.,\({\rm{1993: 1600 - 1621}}\)) suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is\({\rm{.5}}\)year. a. How many loads can be expected to occur during a\({\rm{2}}\)-year period? b. What is the probability that more than five loads occur during a\({\rm{2}}\)-year period? c. How long must a time period be so that the probability of no loads occurring during that period is at most\({\rm{.1}}\)?

The amount of flow through a solenoid valve in an automobile鈥檚

pollution-control system is an important characteristic. An experiment was carried out to study how flow rate depended on three factors: armature length, spring load, and bobbin depth. Two different levels (low and high) of each factor were chosen, and a single observation on flow was made for each combination oflevels.

a. The resulting data set consisted of how manyobservations?

b. Is this an enumerative or analytic study? Explainyour reasoning.

In a famous experiment carried out in 1882, Michelson,and Newcomb obtained 66 observations on the time it took for light to travel between two locations in Washington, D.C. A few of the measurements(coded in a certain manner) were 31, 23, 32, 36, 22, 26, 27, and 31.

a. Why are these measurements not identical?

b. Is this an enumerative study? Why or why not?