In a certain cell population, cells divide every 10 days, and the age of a cell selected at random is a random variable X with the density function f (x) = 2ke-kx, 0 ≤ x ≤ 10, k = (ln 2)/10. Upon examination of a slide, 10% of the cells are found to be undergoing mitosis (a change in the cell leading to division). Compute the length of time required for mitosis; that is, find the number M such that ∫10-M 10 2ke-kx dx = .10.
> The U.S. Energy Information Administration releases figures in their publication, Monthly Energy Review, about the cost of various fuels and electricity. Shown here are the figures for four different items over a 12-year period. Use the data and stepwise
> The National Underwriter Company in Cincinnati, Ohio, publishes property and casualty insurance data. Given here is a portion of the data published. These data include information from the U.S. insurance industry about (1) net income after taxes, (2) div
> It appears that over the past 50 years, the number of farms in the United States declined while the average size of farms increased. The following data provided by the U.S. Department of Agriculture show five-year interval data for U.S. farms. Use these
> Study the output given here from a stepwise multiple regression analysis to predict y from four variables. Comment on the output at each step. Stepwise Selection of Terms
> The computer output given here is the result of a stepwise multiple regression analysis to predict a dependent variable by using six predictor variables. The number of observations was 108. Study the output and discuss the results. How many predictors en
> Given here are data for a dependent variable and four potential predictors. Use these data and a stepwise regression procedure to develop a multiple regression model to predict y. Examine the values of t and R2 at each step and comment on those values. H
> Use a stepwise regression procedure and the following data to develop a multiple regression model to predict y. Discuss the variables that enter at each step, commenting on their t values and on the value of R2.
> A researcher gathered 155 observations on four variables: job satisfaction, occupation, industry, and marital status. She wants to develop a multiple regression model to predict job satisfaction by the other three variables. All three predictor variables
> Falvey, Fried, and Richards developed a multiple regression model to predict the average price of a meal at New Orleans restaurants. The variables explored included such indicator variables as the following: Accepts reservations, Has its own parking lot,
> Given here is Excel output for a multiple regression model that was developed to predict y from two independent variables, x1 and x2. Variable x2 is a dummy variable. Discuss the strength of the multiple regression model on the basis of the output. Focus
> Use the following data to develop a quadratic model to predict y from x. Develop a simple regression model from the data and compare the results of the two models. Does the quadratic model seem to provide any better predictability? Why or why not?
> Using the data in Problem 13.5, develop a multiple regression model to predict per capita personal consumption by the consumption of paper, fish, and gasoline. Discuss the output and pay particular attention to the F test and the t tests. Refer to the P
> Displayed here is the Minitab output for a multiple regression analysis. Study the ANOVA table and the t ratios and use these to discuss the strengths of the regression model and the predictors. Does this model appear to fit the data well? From the infor
> Is it possible to predict the annual number of business bankruptcies by the number of firm births (business starts) in the United States? The following data, published by the U.S. Small Business Administration, Office of Advocacy, are pairs of the number
> Determine the value of the coefficient of correlation, r, for the following data.
> Let X be a continuous random variable with values between A = 1 and B = ∞, and with the density function f (x) = 4x-5. Compute E(X ) and Var (X ).
> The amount of milk (in thousands of gallons) that a dairy sells each week is a random variable X with the density function f (x) = 4(x - 1)3, 1 ≤ x ≤ 2. (See Fig. 4.) (a) What is the likelihood that the dairy will
> When preparing a bid on a large construction project, a contractor analyzes how long each phase of the construction will take. Suppose that the contractor estimates that the time required for the electrical work will be X hundred worker-hours, where X is
> At a certain bus stop the time between buses is a random variable X with the density function f (x) = 6x(10 - x)/1000, 0 ≤ x ≤ 10. Find the average time between buses.
> The amount of time (in minutes) that a person spends reading the editorial page of the newspaper is a random variable with the density function f (x) = 1/72 x, 0 ≤ x ≤ 12. Find the average time spent reading the editorial page.
> The time (in minutes) required to complete an assembly on a production line is a random variable X with the cumulative distribution function F (x) = 1/125 x3, 0 ≤ x ≤ 5. (a) Find E(X ) and give an interpretation of this quantity. (b) Compute Var (X ).
> The useful life (in hundreds of hours) of a certain machine component is a random variable X with the cumulative distribution function F (x) = 1/9 x2, 0 ≤ x ≤ 3. (a) Find E(X ), and give an interpretation of this quantity. (b) Compute Var (X ).
> Let X be the proportion of new restaurants in a given year that make a profit during their first year of operation, and suppose that the density function for X is f (x) = 20x3(1 - x), 0 ≤ x ≤ 1. (a) Find E(X ) and give an interpretation of this quantity
> A newspaper publisher estimates that the proportion X of space devoted to advertising on a given day is a random variable with the beta probability density f (x) = 30x2(1 - x)2, 0 ≤ x ≤ 1. (a) Find the cumulative distribution function for X. (b) Find t
> Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). f (x) = 3√x/16, 0 ≤ x ≤ 4
> The density function f (x) for the lifetime of a certain battery is shown in Fig. 1. Each battery lasts between 3 and 10 hours. (a) Sketch the graph of the corresponding cumulative distribution function F (x). (b) What is the meaning of the number F (7)
> Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). f (x) = 12 x(1 - x)2, 0 ≤ x ≤ 1
> Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). f (x) = 3/2 x – ¾ x2, 0 ≤ x ≤ 2
> Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). f (x) = 5x4, 0 ≤ x ≤ 1
> Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). f (x) = 8/9 x, 0 ≤ x ≤ 32
> Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). f (x) = 1/4, 1 ≤ x ≤ 5
> Verify that each of the following functions is a probability density function. f (x) = 1/18 x, 0 ≤ x ≤ 6
> Find the value of k that makes the given function a probability density function on the specified interval. f (x) = k/√x, 1 ≤ x ≤ 4
> Find the value of k that makes the given function a probability density function on the specified interval. f (x) = k, 5 ≤ x ≤ 20
> Find the value of k that makes the given function a probability density function on the specified interval. f (x) = kx2, 0 ≤ x ≤ 2
> Find the value of k that makes the given function a probability density function on the specified interval. f (x) = kx, 1 ≤ x ≤ 3
> The annual incomes of the households in a certain community range between 5 and 25 thousand dollars. Let X represent the annual income (in thousands of dollars) of a household chosen at random in this community, and suppose that the probability density f
> Verify that each of the following functions is a probability density function. f (x) = 3/2 x – ¾ x2, 0 ≤ x ≤ 2
> Verify that each of the following functions is a probability density function. f (x) = 5x4, 0 ≤ x ≤ 1
> Verify that each of the following functions is a probability density function. f (x) = 8/9 x, 0 ≤ x ≤ 3/2
> Verify that each of the following functions is a probability density function. f (x) = ¼, 1 ≤ x ≤ 5
> Verify that each of the following functions is a probability density function. f (x) = 2(x - 1), 1 ≤ x ≤ 2
> Let X be a continuous random variable with the density function f (x) = 2(x + 1)-3, x ≥ 0. (a) Verify that f (x) is a probability density function for x ≥ 0. (b) Find the cumulative distribution function for X. (c) Compute Pr (1 ≤ X ≤ 2) and Pr (3 ≤ X ).
> Let X be a continuous random variable with values between A = 1 and B = ∞, and with the density function f (x) = 4x-5. (a) Verify that f (x) is a probability density function for x ≥ 1. (b) Find the corresponding cumulative distribution function F (x). (
> A random variable X has a cumulative distribution function F (x) = (x - 1)2 on 1 ≤ x ≤ 2. Find b such that Pr (X ≤ b) = ¼.
> A random variable X has a cumulative distribution function F (x) = 1/4 x2 on 0 ≤ x ≤ 2. Find b such that Pr (X ≤ b) = .09.
> A random variable X has a density function f (x) = 2/3 x on 1 ≤ x ≤ 2. Find a such that Pr (a ≤ X ) = 13.
> A random variable X has a cumulative distribution function F (x) = (x2 - 9)>16 on 3 ≤ x ≤ 5. (a) Find the density function for X. (b) Find a such that Pr (a ≤ X ) = ¼.
> A random variable X has a density function f (x) = 1/3, 0 ≤ x ≤ 3. Find b such that Pr (0 ≤ X ≤ b) = .6.
> In a certain cell population, cells divide every 10 days, and the age of a cell selected at random is a random variable X with the density function f (x) = 2ke-kx, 0 ≤ x ≤ 10, k = (ln 2)/10. Find the probability that a cell is at most 5 days old.
> An experiment consists of selecting a point at random from the region inside the triangle in Fig. 10(b). Let X be the sum of the coordinates of the point. Find the corresponding density function of X. Figure 10(b):
> An experiment consists of selecting a point at random from the region inside the triangle in Fig. 10(b). Let X be the sum of the coordinates of the point. Show that the cumulative distribution function of X is F (x) = x2/4, 0 ≤ x &aci
> An experiment consists of selecting a point at random from the region inside the square in Fig. 10(a). Let X be the maximum of the coordinates of the point. Find the corresponding density function of X. Figure 10(a):
> An experiment consists of selecting a point at random from the region inside the square in Fig. 10(a). Let X be the maximum of the coordinates of the point. Show that the cumulative distribution function of X is F (x) = x2/4, 0 ≤ x &a
> The density function for a continuous random variable X on the interval 1 ≤ x ≤ 4 is f (x) = 4/9 x – 1/9 x2. (a) Use f (x) to compute Pr (3 ≤ X ≤ 4). (b) Find the corresponding cumulative distribution function F (x). (c) Use F (x) to compute Pr (3 ≤ X ≤
> The time (in minutes) required to complete a certain subassembly is a random variable X with the density function f (x) = 1/21 x2, 1 ≤ x ≤ 4. (a) Use f (x) to compute Pr (2 ≤ X ≤ 3). (b) Find the corresponding cumulative distribution function F (x). (c)
> Compute the cumulative distribution function corresponding to the density function f (x) = ½ (3 - x), 1 ≤ x ≤ 3.
> A random variable X has a uniform density function f (x) = 15 on 20 ≤ x ≤ 25. (a) Find E (X ) and Var (X ). (b) Find b such that Pr (X ≤ b) = .3.
> Compute the cumulative distribution function corresponding to the density function f (x) = 1/5 , 2 ≤ x ≤ 7.
> The cumulative distribution function for a random variable X on the interval 1 ≤ x ≤ 2 is F (x) = 4/3 – 4/(3x2). Find the corresponding density function.
> The cumulative distribution function for a random variable X on the interval 1 ≤ x ≤ 5 is F (x) = ½ √(x – 1). (See Fig. 9.) Find the corresponding density function.
> At a certain supermarket, the amount of wait time at the express lane is a random variable with density function f (x) = 11/[10(x + 1)2], 0 ≤ x ≤ 10. (See Fig. 8.) Find the probability of having to wait less than 4
> Suppose that the lifetime X (in hours) of a certain type of flashlight battery is a random variable on the interval 30 ≤ x ≤ 50 with density function f (x) = 1/20, 30 ≤ x ≤ 50. Find the probability that a battery selected at random will last at least 35
> Find Pr (1 ≤ X ) when X is a random variable whose density function is given in Exercise 4. Exercise 4: f (x) = 8/9 x, 0 ≤ x ≤ 3/2
> Find Pr (X ≤ 3) when X is a random variable whose density function is given in Exercise 3. Exercise 3: f (x) = ¼, 1 ≤ x ≤ 5
> Find Pr (1.5 ≤ X ≤ 1.7) when X is a random variable whose density function is given in Exercise 2. Exercise 2: f (x) = 2(x - 1), 1 ≤ x ≤ 2
> Find Pr(1 ≤ X ≤ 2) when X is a random variable whose density function is given in Exercise 1. Exercise 1: f (x) = 1/18 x, 0 ≤ x ≤ 6
> The density function of a continuous random variable X is f (x) = 3x2, 0 ≤ x ≤ 1. Sketch the graph of f (x) and shade in the areas corresponding to (a) Pr (X ≤ .3); (b) Pr (.5 ≤ X ≤ .7); (c) Pr (.8 ≤ X ).
> A service contract on a computer costs $100 per year. The contract covers all necessary maintenance and repairs on the computer. Suppose that the actual cost to the manufacturer for providing this service is a random variable X (measured in hundreds of d
> The density function of a continuous random variable X is f (x) = 1/8 x, 0 ≤ x ≤ 4. Sketch the graph of f (x) and shade in the areas corresponding to (a) Pr (X ≤ 1); (b) Pr (2 ≤ X ≤ 2.5); (c) Pr (3.5 ≤ X ).
> Find the value of k that makes the given function a probability density function on the specified interval. f (x) = k(3x - x2), 0 ≤ x ≤ 3
> Find the value of k that makes the given function a probability density function on the specified interval. f (x) = kx2(1 - x), 0 ≤ x ≤ 1
> Suppose that the weather forecast in Exercise 9 indicates a 10% chance that cold weather will reduce the citrus grower’s profit from $100,000 to $85,000 and a 10% chance that cold weather will reduce the profit to $75,000. Should the grower spend $5000 t
> A citrus grower anticipates a profit of $100,000 this year if the nightly temperatures remain mild. Unfortunately, the weather forecast indicates a 25% chance that the temperatures will drop below freezing during the next week. Such freezing weather will
> Consider a circle with circumference 1. An arrow (or spinner) is attached at the center so that, when flicked, it spins freely. Upon stopping, it points to a particular point on the circumference of the circle. Determine the likelihood that the point is
> Consider a circle with radius 1. (a) What percentage of the points lies within ½ unit of the center? (b) Let c be a constant with 0 < c < 1. What percentage of the points lies within c unit of the center?
> The number of phone calls coming into a telephone switchboard during each minute was recorded during an entire hour. During 30 of the 1-minute intervals there were no calls, during 20 intervals there was one call, and during 10 intervals there were two c
> The number of accidents per week at a busy intersection was recorded for a year. There were 11 weeks with no accidents, 26 weeks with one accident, 13 weeks with two accidents, and 2 weeks with three accidents. A week is to be selected at random and the
> Compute the variances of the two random variables whose probability tables are given in Table 7. Relate the sizes of the variances to the spread of the values of the random variables. Table 7:
> A certain gas station sells X thousand gallons of gas each week. Suppose that the cumulative distribution function for X is F (x) = 1 – ¼ (2 - x)2, 0 ≤ x ≤ 2. (a) If the tank contains 1.6 thousand gallons at the beginning of the week, find the probabilit
> Compute the variances of the three random variables whose probability tables are given in Table 6. Relate the sizes of the variances to the spread of the values of the random variable. Table 6:
> Find E(X ), Var(X ), and the standard deviation of X, where X is the random variable whose probability table is given in Table 5. Table 5:
> Table 4 is the probability table for a random variable X. Find E(X ), Var(X ), and the standard deviation of X. Table 4:
> For any positive integer n, the function fn(x) = cn x(n-2)/2 e-x/2, x ≥ 0, where cn is an appropriate constant, is called the chi-square density function with n degrees of freedom. Find c2 and c4 such that f2 (x) and f4 (x) are probability density functi
> For any positive constants k and A, verify that the function f (x) = kAk/xk+1, x ≥ A, is a density function. The associated cumulative distribution function F (x) is called a Pareto distribution. Compute F (x).
> For any number A, verify that f (x) = eA-x, x ≥ A, is a density function. Compute the associated cumulative distribution for X.
> Let X be a continuous random variable on 3 ≤ x ≤ 4, with the density function f (x) = 2(x - 3). (a) Calculate Pr (3.2 ≤ X ) and Pr (3 ≤ X ). (b) Find E(X ) and Var(X ).
> Let X be a continuous random variable on 0 ≤ x ≤ 2, with the density function f (x) = 3/8 x2. (a) Calculate Pr (X ≤ 1) and Pr (1 ≤ X ≤ 1.5). (b) Find E(X ) and Var (X ).
> A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9. What is the probability that at least three consecutive rolls precede the final roll?
> A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9. Determine the average number of consecutive rolls preceding the final roll.
> If the laboratory in Exercise 7 uses batches of 5 instead of 10 samples, the probability of a negative test on the mixture of 5 samples is (.95)5 = .774. Thus, Table 2 gives the probabilities for the number X of tests required. (a) Find E(X ). (b) If the
> A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9. Determine the formula for pn, the probability of exactly n consecutive rolls preceding the final
> A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen is Poisson distributed with λ = 4. What is the average
> A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen is Poisson distributed with λ = 4. What is the probabi
> A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen is Poisson distributed with λ = 4. What is the probabi
> Do the same as in Exercise 29 with a normal random variable. Exercise 29: The Chebyshev inequality says that for any random variable X with expected value m and standard deviation σ, Pr (µ - nσ ≤ X ≤ µ + nσ) ≥ 1 -1/n2. (a) Take n = 2. Apply the Chebyshe
> The Chebyshev inequality says that for any random variable X with expected value m and standard deviation σ, Pr (µ - nσ ≤ X ≤ µ + nσ) ≥ 1 -1/n2. (a) Take n = 2. Apply the Chebyshev inequality to an exponential random variable. (b) By integrating, find th
> (a) Show that about 95% of the area under the standard normal curve lies between -2 and 2. (b) Let X be a normal random variable with expected value µ and variance σ2. Compute Pr (µ - 2σ ≤ X ≤ µ + 2σ).
> It is useful in some applications to know that about 68% of the area under the standard normal curve lies between -1 and 1. (a) Verify this statement. (b) Let X be a normal random variable with expected value µ and variance σ2. Compute Pr (µ - σ ≤ X ≤ µ
> Scores on an entrance exam Scores on a school’s entrance exam are normally distributed, with µ = 500 and σ = 100. If the school wishes to admit only the students in the top 40%, what should be the cutoff grade?
