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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

7.82 What survey design is used in each of these situations?a. A random sample of n = 50 city blocks is selected, and a census is done for each single-family dwelling on each block.b. The highway
7.83 Elevator Loads The maximum load (with a generous safety factor) for the elevator in an office building is 900 kg. The relative frequency distribution of the weights of all men and women using
7.84 Wiring Packages The number of wiring pack- ages that can be assembled by a company's employees has a normal distribution, with a mean equal to 16.4 per hour and a standard deviation of 1.3 per
7.85 Wiring Packages, continued Refer to Exercise 7.84. Suppose the company employs 10 assemblers of wiring packages.a. Find the mean and standard deviation of the com- pany's daily (8-hour day)
7.86 Defective Light bulbs The table lists the number of defective 60-watt light bulbs found in samples of 100 bulbs selected over 25 days from a manufacturing process. Assume that during these 25
Where do Canada's most clever people live? Who wins the IQ battle of the sexes? We found out on March 18, 2007. "Test the Nation" was the biggest survey ever conducted to see just how smart Ca-
A computer database at a downtown law firm contains files for N = 1000 clients. The firm wants to select n = 5 files for review. Select a simple random sample of 5 files from this database.
7.13 Tai Chi and Fibromyalgia A small new study shows that tai chi, an ancient Chinese practice of exer- cise and meditation, may relieve symptoms of chronic painful fibromyalgia. The study assigned
7.14 Blood Thinner A study of an experimental blood thinner was conducted to determine whether it works better than the simple aspirin tablet in warding off heart attacks and strokes.' The study
7.15 Health Care: Canada Speaks Two differ- ent polls were conducted by two different organizations, both of which involved people's feelings about national priorities/important issues. Here is a
7.16 Ask Canada A nationwide policy survey was sent by the Conservative Party Election Committee to voters asking for opinions on a variety of political issues. Here are some questions from the
A population consists of N = 5 numbers: 3, 6, 9, 12, 15. If a random sample of size n = 3 is selected without replacement, find the sampling distributions for the sample mean x and the sample median
The duration of Alzheimer's disease from the onset of symptoms until death ranges from 3 to 20 years; the average is 8 years with a standard deviation of 4 years. The administrator of a large medical
To avoid difficulties with the federal or provincial and local consumer protection agen- cies, a beverage bottler must make reasonably certain that 355 millilitre (mL) bottles actually contain 355 mL
Statistics Canada reports that the birth weight of newborn babies in Saskatchewan has a mean of 3.45 kg for both sexes." Suppose the standard deviation is 0.70 kg. Further, we randomly sample 49
7.17 Random samples of size n were selected from populations with the means and variances given here. Find the mean and standard deviation of the sampling distribution of the sample mean in each
7.19 Refer to Exercise 7.17, partb. a. Sketch the sampling distribution for the sample mean and locate the mean and the interval 20/Vn along the x-axis.b. Shade the area under the curve that
7.20 A population consists of N = 5 numbers: 1, 3, 5, 6, and 7. It can be shown that the mean and standard deviation for this population are = 4.4 and 2.15, respectively.a. Construct a probability
7.21 Refer to Exercise 7.20.a. Use the data entry method in your calculator to find the mean and standard deviation of the 50 values of x given in Exercise 7.20, partc. b. Compare the values
7.12 MRIs In a study described in the American Journal of Sports Medicine, Peter D. Franklin and col- leagues reported on the accuracy of using magnetic resonance imaging (MRI) to evaluate ligament
7.11 Racial Bias? Does the race of an interviewer matter? This question was investigated by Chris Gilberg and colleagues and reported in an issue of Chance magazine. The interviewer asked, "Do you
A research chemist is testing a new method for measuring the amount of titanium (Ti) in ore samples. She chooses 10 ore samples of the same weight for her experi- ment. Five of the samples will be
Identify the sampling design for each of the following: 1. The faculty of mathematics and science at the Brock University consists of six academic departments and two centres. Dr. S. Ejaz Ahmed, the
7.1 A population consists of N = 500 experimental units. Use a random number table to select a random sample of n = 20 experimental units. (HINT: Since you need to use three-digit numbers, you can
7.5 Every 10th Person A random sample of public opinion in a small town was obtained by selecting every 10th person who passed by the busiest corner in the down- town area. Will this sample have the
7.6 Parks and Recreation A questionnaire was mailed to 1000 registered municipal voters selected at random. Only 500 questionnaires were returned, and of the 500 returned, 360 respondents were
7.7 MPAC Lists and Jury Selection Juries are selected in Ontario through a process described in the Juries Act. Enumeration lists are obtained from the Municipal Property Assessment Corporation
7.8 Sex and Violence One question on a survey questionnaire is phrased as follows: "Don't you agree that there is too much sex and violence during prime TV viewing hours?" Comment on possible
7.9 Omega-3 Fats Contrary to current thought about omega-3 fatty acids, new research shows that the ben- eficial fats may not help reduce second heart attacks in heart attack survivors. The study
7.10 Cancer in Rats The Press Enterprise identified a byproduct of chlorination called MX that has been linked to cancer in rats. A scientist wants to conduct a valida- tion study using 25 rats in
7.22 A random sample of n observations is selected from a population with standard deviation = 1. Calculate the standard error of the mean (SE) for these values of n: a. n 1 b. n = 2 c. n=4 d. n = 9
7.47 Calculate SE(p) for n = 100 and these values of p:a. p = 0.01b. p = 0.10c. p = 0.30d. p = 0.50e. p = 0.70f. p = 0.90 g. p = 0.99 h. Plot SE(p) versus p on graph paper and sketch a smooth curve
7.37 Potassium Levels The normal daily human potassium requirement is in the range of 2000 to 6000 milligrams (mg), with larger amounts required during hot summer weather. The amount of potassium in
7.38 Deli Sales The total daily sales, x, in the deli section of a local market is the sum of the sales generated by a fixed number of customers who make purchases on a given day.a. What kind of
7.39 Normal Temperatures In Exercise 1.68, Allen Shoemaker derived a distribution of human body temperatures with a distinct mound shape.10 Suppose we assume that the temperatures of healthy humans
7.40 Sports and Achilles Tendon Injuries Some sports that involve a significant amount of running, jumping, or hopping put participants at risk for Achilles tendinopathy (AT), an inflammation and
A sports facility located in Prince Edward Island recently conducted a survey on the importance of sports for children. In the survey, 500 mothers and fathers were asked about the importance of
Refer to Example 7.8. Suppose the proportion p of parents in the population is actually equal to 0.55. What is the probability of observing a sample proportion as large as or larger than the observed
7.41 Random samples of size n were selected from binomial populations with population parameters p given here. Find the mean and the standard deviation of the sampling distribution of the sample
7.42 Sketch each of the sampling distributions in Exercise 7.41. For each, locate the mean p and the interval p 2 SE along the p-axis of the graph.
7.43 Refer to the sampling distribution in Exercise 7.41, parta. a. Sketch the sampling distribution for the sample proportion and shade the area under the curve that corresponds to the probability
7.44 Is it appropriate to use the normal distribution to approximate the sampling distribution of p in the fol- lowing circumstances?a. n = 50, p = 0.05b. n = 75, p = 0.1c. n=250, p = 0.99
7.45 Random samples of size n = 75 were selected from a binomial population with p = 0.4. Use the normal distribution to approximate the following probabilities:a. P(p 0.43)b. P(0.35 p 0.43)
7.46 Random samples of size n = 500 were selected from a binomial population with p = 0.1.a. Is it appropriate to use the normal distribution to approximate the sampling distribution of p? Check to
7.24 A random sample of n observations is selected from a population with standard deviation = 5. Calculate the standard error of the mean (SE) for these values of n:a. n = 1b. n = 2c. n = 4d. n =
7.36 Paper Strength A manufacturer of paper used for packaging requires a minimum strength of 1400 g/cm. To check on the quality of the paper, a random sample of 10 pieces of paper is selected each
7.23 Refer to Exercise 7.22. Plot the standard error of the mean (SE) versus the sample size n and con- nect the points with a smooth curve. What is the effect of increasing the sample size on the
7.25 Refer to Exercise 7.24. Plot the standard error of the mean (SE) versus the sample size n and connect the points with a smooth curve. What is the effect of increasing the sample size on the
7.26 A random sample of size n = 49 is selected from a population with mean = 53 and standard deviation = 21.a. What will be the approximate shape of the sampling distribution of X?b. What will be
7.27 Refer to Exercise 7.26. Find the probability that the sample mean is greater than 55.
7.28 A random sample of size n = 40 is selected from a population with mean deviation or = 20. = 100 and standarda. What will be the approximate shape of the sampling distribution of X?b. What will
7.29 Refer to Exercise 7.28. Find the probability that the sample mean is between 105 and 110.
7.31 Measurement Error When research chemists perform experiments, they may obtain slightly dif- ferent results on different replications, even when the experiment is performed identically each time.
7.32 Tomatoes Explain why the weight of a pack- age of one dozen tomatoes should be approximately normally distributed if the dozen tomatoes represent a random sample.
7.33 Bacteria in Water Use the Central Limit Theorem to explain why a Poisson random variable- say, the number of a particular type of bacteria in a cubic metre of water-has a distribution that can
7.34 Faculty Salaries Suppose that university faculty with the rank of assistant professor earn an average of $74,000 per year with a standard deviation of $6000. In an attempt to verify this salary
7.35 Tax Savings An important expectation of a federal income tax reduction is that consumers will reap a substantial portion of the tax savings. Suppose estimates of the portion of total tax saved,
The density function of X is f (x) = 32x−3, x ≥ 4.The expected value of X is 2.
If the distribution of X has a jump at the pointa, then P(X =a) = 0.
If X has a mixed distribution, then its range contains infinitely many points.
The distribution function of X is given by F(x) =⎧⎪⎨⎪⎩0, x < 0(x2 + 1)∕5, 0 ≤ x < 2 1, x ≥ 2.Then X has a mixed distribution.
If a continuous random variable X has distribution function F, then for anya, b, P(a < X
A random variable X has density f (x) = 2(3x − x2)9 , 0 < x < 3.Then the probability P(1 < X < 2) equals 13∕27.
The manufacturing time, X, for an item produced by a machine has distribution function F(t) = 1 − e−2t2, t ≥ 0.Then, the probability P(X > 2) equals e−8.16. A random variable has probability
Let X be a continuous variable whose range is the entire real line and which has distribution function FX, and let Y = |X|. Then for the distribution function, FY , of Y, we have FY (y) = FX(y) −
Let X be a random variable with density function f (x) = 1 3, 0 ≤ x ≤ 3.Then we have P(X2 ≤ 1) = 1∕3.
If X has distribution function F(t) =⎧⎪⎨⎪⎩0, t < 2,(t − 2)∕4, 2 ≤ t < 6, 1, t ≥ 6, then E(X) = 5.
The density function of a random variable X is f (x) = xe−x, x > 0.The distribution function of X (for x > 0) is(a) 1 − e−x (b) 1 − xe−x (c) 1 − x − e−x(d) 1 − (1 + x)e−x (e) 1
The density function of X is f (x) = c(x + 3)2 , 0 ≤ x ≤ 3.Then the value of c is(a) 2 (b) 6 (c) 3 (d) 1∕6 (e)1∕2
A random variable X has distribution function F(t) =⎧⎪⎨⎪⎩0, t < −2,(t + 2)∕8, −2 ≤ t < 1, 5∕8, 1 ≤ t < 3, 1, t ≥ 3.Then the probability P(−1 ≤ X < 3) equals(a) 5 8(b) 3
A continuous variable X has density function f (x) = 3x2, 0 ≤ x ≤ b.The value of b is(a) 1 (b) 1∕3 (c) 3 (d)3 √3 (e) 3 √2
A random variable has probability density function f (x) = c|x|, −2 ≤ x ≤ 2.The value of c is(a) 1 (b) 2 (c) 4 (d) 1∕2 (e)1∕4
Let X be a continuous variable with density fX(x) = (2x + 1)∕6, 0 < x < 2.Then the density function of Y = X2 is fY (y) = (2y + 1)2∕36, 0 < y < 4.
Let f be a density function of a random variable X. Then, we have limx→∞ f (x) = 1.
The function f (x) = 3(4x − x2)∕8, 0 ≤ x ≤ 4, can be the density function of a continuous random variable.
Let FX be the distribution function and fX be the density function of a variable X for which we know that P(X ≥a) = 1, where a ∈ ℝ is a given constant. We define the functionwhere 0 (i) Verify
A random variable X has distribution functionwhere ???? and m are positive constants.(i) Identify the point(s) at which F has jumps.(ii) Find the density function associated with the continuous part
The lifetime of an electrical appliance (in thousands of hours) is a random variable X with density function f (x) ={0, x < 0, 0.5 ⋅ (x∕20)9 ⋅ e−(x∕20)10, x ≥ 0.(i) Find the distribution
Draw a graph of the density functions for the distributions given in Exercise 21 of Section 6.1. Hence find, in each case, the point a around which the density is symmetric. Moreover, verify
Suppose that X is a continuous random variable with distribution function⎧⎪⎨⎪⎩0, t < 2, 0.5t − 1, 2 ≤ t ≤ 4, 1, t > 4.Draw a graph of the difference between the exact value of the
Find the value of the constant c ∈ ℝ for which the function f defined by f (x) =⎧⎪⎨⎪⎩c ⋅2 + cos(√x)ex∕3 , 0 ≤ x ≤ π2, 0, elsewhere, is a density function for a continuous
The value of an investment at the end of a certain period is described by a random variable X with density function f (x) =⎧⎪⎨⎪⎩1 x√2πe−(ln x−4)2∕2, x > 0, 0, x ≤ 0.(i) Find the
The density function of a random variable X is given by f (x) = c ⋅ e−|x|, −∞ < x < ∞, for a real constant c.(i) Find the value of c and draw the graphs of the density function and the
After reading the applications section of this chapter (Section 6.10), draw a graph of the mean monthly profit, as a function of the ordered quantity z, and find the value of z that maximizes this
A probability density function can take positive, zero, and negative values.
A continuous variable X has density function f (x) = 2c + 3, 2 ≤ x ≤ 6.Then the value of c is 1∕2.
The density function of a random variable is always a decreasing function.
If a continuous random variable X with density function f and distribution function F takes only positive values then, for any x > 0, we have F(x) =∫x 0f (y)dy.
For a continuous random variable X with distribution function F, we have F′(x) =P(X = x).
An insurance company classifies the claims it receives as being either small (if they are up to $5000) or large. During a calendar year, an insured customer may either make one claim or none. It has
The density function of a random variable is given by f (x) = 2x(c + 1)5 , 0 < x < 2.The distribution function of X is(a) F(t) =⎧⎪⎨⎪⎩0, t < 0, t∕2, 0 ≤ t < 2, 1, t ≥ 2(b) F(t)
The pressure P, in lb ft−2, developed at the wings of an aircraft is given by P = 3 ⋅ 10−3V2, where V is the velocity (in miles per hour) of the wind surrounding the wings. The pressure can be
The measurement error of an instrument can be described by a random variable X with distribution function(i) Calculate the following probabilities P(X (ii) Find the density function and the
The percentage concentration, in alcohol, of a medical substance is a continuous variable X with density function f (x) = cx(1 − x)n, 0 < x < 1, where c is a real constant and n is a positive
The time, in minutes, that a medicine for pain relief takes until it starts to have effect is a random variable X with density functionwhere ???? > 0 is the parameter of the distribution of X.
A continuous random variable X has distribution function(i) Obtain an expression for the probability P(a b) for a 0.(ii) Verify that the density function of X is given byiii) Calculate the expected
Let X be a random variable having the Pareto distribution (Example 6.6). The density of X is given by(i) What is the distribution function F(t) of X?(ii) For a given positive integer n, show
The lifetime, in hundreds of hours, of a light bulb is a continuous random variable with density function f (x) = ????2xe−????x, x > 0, where ???? > 0.(i) Calculate the distribution function of
Let X1 and X2 be two continuous random variables with density functions f1 and f2, expected values ????1 and ????2 and variances ????2 1 and ????2 2 , respectively. Consider now a function f defined
Let X and Y be two continuous random variables with densities fX and fY , respectively, and with the same range of values, R. Is it possible that fX(x) < fY (x) for all x ∈ R?If it is, give an

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