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Statistics And Probability With Applications For Engineers And Scientists Using MINITAB R And JMP 2nd Edition Irwin Guttman, Kalanka P. Jayalath, Bhisham C. Gupta - Solutions
Assuming that a Pearson r can be used to analyze the data in Exercise 13.74, list the hypotheses.
A psychodynamic psychologist used an interval-level scale to measure the degree of Oedipal conflict in a large sample of elementary age boys. He then had each one, by himself, play a target-shooting game where the boy had equal opportunity to shoot at male and female targets. The psychologist
A philosopher wanted to see if curiosity about the meaning of life had any correlation, either positive or negative, with the age of college students and planned to use a Pearson r on her data. She obtained a random sample of 250 undergraduate students at the college where she taught and
“A kindergarten teacher classifies students as coloring inside the lines or coloring outside the lines. Years later, he uses police records to determine how many driving violations each former student has had. Is there a relationship between following the rules in kindergarten and following them
“A behavioral therapist had patients with spider phobias rate the level of their fear on an interval-level scale. He then asked the spider phobics to enter a room with a spider in a cage and come as close to the spider as they felt comfortable. He measured the distance in feet. Do people with
“People are measured, on an interval scale, to determine how fairskinned they are. A dermatologist then examines them to see how many suspicious moles they have in order to determine if fairness of skin relates to the number of suspicious moles.” What statistical test should she use to answer
“People who wear hats in the wintertime are compared to those who don’t wear hats in terms of how many days they suffer from a headcold. Does wearing a hat make a difference?” What test should be used to decide?
Given these data, make a scatterplot:Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 X 44 46 48 47 49 50 48 49 52 52 53 54 54 55 56 Y 29 25 28 34 26 30 32 36 27 37 24 31 34 25 30
Given these data, make a scatterplot:Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 X 5 8 12 16 19 22 26 31 12 25 18 6 30 22 9 Y 5 10 13 12 9 4 5 11 13 6 10 6 8 6 8
A partial correlation mathematically ___ the effect of a third variable.
Statisticians like power to be at least ___.
(a) If r = – .34, what is r 2? (b) Is this a small, medium, or large effect?
Which figure or figures have a negative linear relationship?
Which figure or figures have a strong, but not perfect, linear relationship?
Which figure or figures have a weak linear relationship?
Which figure or figures represent no relationship?
Which figure or figures have a linear relationship?
Make a scatterplot for these data: X Y X Y 100 110 80 95 90 85 100 95 85 95 110 115 90 95 85 80 80 85 90 105 110 125
Given R = 3, C = 3, n = 12, and the ANOVA summary table below, calculate HSD values for the effects as necessary and appropriate.
The means of six independent samples are being compared.Which ANOVA is being used?a. between-subjects, one-way ANOVAb. one-way, repeated-measures ANOVAc. between-subjects, two-way ANOVAd. (a) or (b)e. (a) or (c)f. not enough information provided to decide
The means of five independent samples are being compared.Which ANOVA is being used?a. between-subjects, one-way ANOVAb. one-way, repeated-measures ANOVAc. between-subjects, two-way ANOVA
Each cell in the table has five cases. (a) Given the cell, row, and column means below, calculate the missing cell means. If it can’t be done, say so. (b) Calculate the mean for all 30 cases. If it can’t be done, say so.
Each cell in this table has the same number of cases, but that number is unknown. (a) Given the cell, row, and column means below, calculate the missing cell means. If it can’t be done, say so.(b) Calculate the mean for all the cases. If it can’t be done, say so.
Each cell in the table contains seven cases. (a) Given the cell, row, and column means below, calculate the missing cell means. If it can’t be done, say so. (b) Calculate the mean for all 28 cases. If it can’t be done, say so.
A developmental psychologist investigated the influence of two crossed variables—exposure to televised violence and type of parental discipline—on teens’ acceptance of violence. He obtained a random sample of seniors at the local high school and classified them on two dimensions: (1) whether
A cognitive psychologist decided to investigate whether two beliefs about healthy living had any real impact on performance in college.She took 40, first-semester college student volunteers and randomly assigned half to eat breakfast every day and the other half to skip breakfast every day. This
A consumer psychologist classified shoppers at a grocery store as(a) being males or females, and (b) shopping with or without children. These two variables were crossed and he took a random sample of five shoppers from each of the four cells. Then he gave them the Enjoyment of Shopping Experience
If MCell 1 = 55.54, MCell 2 =48.34, MCell 3 =36.44, MCell 4= 59.40, and HSDCells= 7.83, determine (a) which cells have a statistically significant difference and (b) the direction of the differences.
If MRow 1 = 117.66, MRow 2 = 113.63, MRow 3 = 128.91, and HSDRows = 5.89, determine (a) which rows have a statistically significant difference and (b) the direction of the differences.(p. 18)
If α = .05, for which effects should an HSD value be calculated? Source of Variability Sum of Squares Degrees of Freedom Mean Square Fratio Between groups 118.00 8 Rows 88.00 2 44.00 9.91 Columns 23.00 2 11.50 2.59 Interaction 7.00 4 1.75 0.39 Within groups 600.00 135 4.44
If α = .05, for which effects should an HSD value be calculated? Source of Variability Sum of Squares Degrees of Freedom Mean Square F ratio Between groups 324.00 8 Rows 70.00 2 35.00 28.46 Columns 88.00 2 44.00 35.77 Interaction 166.00 4 41.50 33.74 Within groups 100.00 81 1.23 Total 424.00 89
If qCells = 2.96, MSWithin = 12.75, and nCells = 4, what is HSDCells?(p. 18)
If qRows = 3.49, MSWithin = 4.56, and nRows = 12, what is HSDRows?(p. 18)
Given the ANOVA summary table below, calculate n 2for (a) the rows effect, (b) the columns effect, and (c) the interaction effect.
If the null hypothesis is rejected for all three effects (rows, columns, and interaction), which effects should be interpreted?
Given the ANOVA summary table below, write the results in APA format for (a) the rows effect, (b) the columns effect, and (c) the interaction effect. Use a = .05. Source of Variability Sum of Squares Degrees of Freedom Mean Square Fratio Between groups 70.40 7 Rows 32.34 3 10.78 2.77 Columns 7.62 1
List all the hypotheses for the ANOVA described in Exercise 12.53 for (a) rows, (b) columns, and (c) interaction.
To study accommodation to the loss of an eye, a sensory psychologist obtained 10 volunteers who agreed to wear an eye patch over one eye for 4 weeks. To test visual ability, the psychologist used a batting test: a pitching machine lobbed 50 pitches to each participant and the psychologist counted
Male and female athletes exercised for an hour on a treadmill.During this time, half of each sex was assigned to drink water and half was assigned to drink a sport beverage. At the end of the hour, lactic acid levels were measured and mean levels were calculated.What statistical test should be used
A dentist randomly assigns people to use one of three different forms of dental hygiene: (1) dental floss, (2) wooden toothpicks, or(3) anti-plaque rinse. After six months he measures, in grams, how much plaque he scrapes off the teeth. What statistical test should he use to see if form of dental
Post-hoc tests should only be used when an effect is____.
If the lines in a graph are____, then an interaction exists.(p. 18)
An interaction occurs when the effect of one explanatory variable on the dependent variable ___on the level of the other explanatory variable.(p. 18)
A two-way ANOVA has two___effects and one___effect.(p. 18)
If every level of one explanatory variable occurs with every level of the other explanatory variable, then the two explanatory variables are said to be___.
There is an advantage in performing one ___-way ANOVA over two___-way ANOVAs.(p. 18)
Two-way ANOVA, three-way ANOVA, and four-way ANOVA are all examples of___ ANOVA.(p. 18)
“Between subjects” means___samples and “___subjects” means dependent samples.(p. 18)
A 2 × 3 ANOVA could also be called a___× ___ANOVA.(p. 18)
____ -way ANOVA examines the impact of two explanatory variables at the same time.
A kinesiologist wanted to investigate the effect of temperature and humidity on human performance. He found 28 college students and randomly assigned them to four different conditions, during which they were to walk at their normal pace on a treadmill for 60 minutes.He measured how far, in miles,
Here is a table of cell means for a 3 × 2 between-subjects, two-way ANOVA with seven cases in each cell. Note that the row means and column means have been calculated.(p. 18) Column 1 Column 2 Column 3 Row 1 26.00 30.00 34.00 30.00 Row 2 18.00 22.00 26.00 22.00 22.00 26.00 30.00
Given this ANOVA summary table, (a) calculate η2 for each effect and (b) classify each effect as small, medium, or large. Use α = .05.(p. 18) Source of Variability Sum of Squares Degrees of Freedom Mean Square F ratio Between groups 5,357.00 15 Rows 3,725.00 3 1,241.67 37.63 Columns 1,312.00 3
Given α = .05, dfRows = 3, dfColumns = 2, df Interaction = 6, dfWithin = 60, FRows = 3.25, FColumns = 1.22, and FInteraction = 0.83, (a) write each result for this between-subjects, two-way ANOVA in APA format and (b) for each result report whether the effect is statistically significant.(p. 18)
Given this ANOVA summary table, (a) calculate η2 for each effect and (b) classify each effect as small, medium, or large. Use α = .05.(p. 18)
Given α = .05, dfRows = 3, dfColumns = 2, df Interaction = 6, dfWithin = 60, FRows = 3.25, FColumns = 1.22, and FInteraction = 0.83, (a) write each result for this between-subjects, two-way ANOVA in APA format and (b) for each result report whether the effect is statistically significant.(p. 18)
Given n = 7, R = 2, C = 2, SSBetween = 650.00, SSRows = 250.00, SSColumns = 300.00, SSInturaction = 100.00, SSWithin = 800.00, and SSTotal = 1,450.00, complete an ANOVA summary table for a between-subjects, two-way ANOVA.(p. 18)
Given dfRows = 2, dfColumns = 4, df Interaction = 8, and dfWithin = 165, list the critical values of F for the three F ratios for a betweensubjects, two-way ANOVA for α = .05.(p. 18)
List the hypotheses for a between-subjects, two-way ANOVA in which there are four rows and three columns.(p. 18)
Read each scenario and decide what statistical test should be used.Select from a single-sample z test; single-sample t test; independentsamples t test; paired-samples t test; between-subjects, one-way ANOVA; one-way, repeated-measures ANOVA; and betweensubjects, two-way ANOVA.a. People who are
Thirty-six cases were randomly divided into four samples. Each person took either a low dose or a high dose of a drug and was queried about either the physical side effects or psychological side effects. Each cell in the matrix below reports the mean number of side effects reported by a sample of
Each cell in the matrix below reports the mean of six cases. (a)Calculate the row and column means. (b) Interpret the two main effects.(p. 18)
Classify each scenario in terms of the number of ways and the number of levels each way has. For example, classify each scenario as a 2 × 3 design, a 2 × 3 × 5 design, or some other variation.a. Men and women who are right-handed or left-handed and who use razors with blades vs. electric razors
26. Referring to Problem 25,(a) Find the conditional p.d.f. of the random variable Y , given X = x.(b) Find the conditional p.d.f. of the random variable X, given Y = y.(c) Find the conditional mean and variance of the random variable Y , given X = x.(d) Find the conditional mean and variance of
25. Suppose that the random variables (X, Y ) are bivariate normal with joint p.d.f.f(x, y) = c exp−25 18x2 + y2 +4 5x − 14 5 y − 8 5xy +17 5(a) Find the parameters of the bivariate normal(b) Find the value of c(c) Find the marginal p.d.f.’s of both X and Y .
24. Suppose that the random variables (X1,X2) are distributed as bivariate normal with parameters 0, 0, 1, 1, and ρ. Show that the random variables X = X1 + X2 and Y =X1 − X2 are independent.
23. Let F(x, y) be the c.d.f. of random variables (X, Y ). Show that P(x1 < X < x2, y1 < Y < y2) = F(x2, y2) − F(x2, y1) − F(x1, y2) + F(x1, y1) ≥ 0
22. Let X1 be a number taken at random on the interval (0, 1), and suppose that X1 = x1 is the observed value of X1. Let X2 be a number taken at random on the interval(x1, 1). Show that the distribution of X2 has p.d.f.f2(x2) =−ln(1 − x2), if 0 < x2 ≤ 1 0, otherwise
21. If (X1,X2) is a pair of random variables such that p2(x2) = μx2e−μx2! , x2 = 0, 1, 2, . . .and p(x1|x2) =x2 x1px1(1 − p)x2−x1, x1 = 0, 1, 2, . . . , x2 show that p1(x1) is a Poisson distribution.
20. Suppose that X1, . . . , Xn is a sample from a distribution whose mean is μ, variance is σ2, and whose moment-generating function exists. Show, using the method of moment-generating functions, that the mean and variance of the sample sum T are nμand nσ2, respectively. Also use the method of
19. By using the moment-generating function of a random variable X having the binomial distribution with parameter p, show that the mean and the variance of X are np and np(1 − p), respectively.252 6 Distribution of Functions of Random Variables
18. A point is taken at random from the interval (0, 1), all points being equally likely.A second point is then taken in the same way. Let X be the coordinate of the point halfway between these points. X is a continuous chance quantity with a probability density function having an inverted V graph
17. One cigarette from each of four brands A, B, C, D is partially smoked by a blindfolded person. As soon as he takes a few puffs on a cigarette, he states the letter of the brand to which he considers it to belong. (Of course, he can use each letter only once.) Let X be the random variable
16. If 2500 coins in a sack are poured out on a table, find with the use of the Chebyshev’s inequality the value of k for which the probability that the number of heads will lie in the interval (1250 − k, 1250 + k) exceeds 0.96.
15. Suppose that an insurance company has among all of its insurance policies, 50,000 policies for $5000 for American men aged 51. The probability of an American male aged 51 dying within one year is (approximately) 0.01. Using the Chebyshev’s inequality, decide for what value of k the
14. Using the Chebyshev’s inequality, determine how many random digits should be generated in order for the probability to exceed 0.9000 that the mean of the random digits will lie within the interval (3.5, 5.5).
13. A sample of size n is drawn from a lot of 10,000 articles known to contain 100 defectives.Using the Chebyshev’s inequality, determine how large n should be in order for the probability to exceed 0.96 that the percentage of defectives in the sample will lie within the interval (0.1, 1.9).
12. A person plays 10 hands of bridge during the evening; suppose that T represents the total number of spades he obtains during the evening. Find the mean and variance of T.
11. A process randomly generates digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 with equal probabilities.If T is a random variable representing the sum of n digits taken from the process, find the mean and the variance of T.Review Practice Problems 251
10. Suppose that 420 “true” dice are rolled simultaneously.(a) If X is a random variable denoting the total number of aces that turn up, find the values of the mean and standard deviation of X.(b) If Y is a random variable denoting the total number of dots that turn up, find the values of the
9. A certain type of half-inch rivet is classified as acceptable by a consumer if its diameter lies between 0.4950 and 0.5050 in. It is known that a mass-production process is such that 100p1% of the rivets have diameters less than 0.4950 in., 100p2% have diameters that lie in the “acceptable”
8. Mass-produced articles are fitted into cardboard containers, one article to a container.Twelve of these filled containers are then packed in wooden boxes. Suppose that the mean and standard deviation of the weights in pounds (lb) of the population of articles are 20.6 and 0.8 lb respectively,
7. A laminated strip is built up by randomly selecting two layers of material A, three layers of material B, and four layers of material C. The thicknesses of the individual layers of material A have mean 0.0100 in., and standard deviation 0.0005 in.; the respective numbers for material B are
6. In packaging corn flakes into 8-oz packages, assume that the population of net weights generated by the automatic filling machine (properly calibrated) has a distribution with mean of 8.15 oz and standard deviation of 0.08 oz. Assume that the population of paper boxes to receive the fillings to
5. A resistor is composed of two component parts soldered together in series; the total resistance of the resistor equals the sum of the resistances of the component parts.The first part is drawn from a production lot having a mean of 200 Ω and standard deviation of 2 Ω, and the second part is
4. Suppose that X1, . . . , Xn are independently distributed with N(μi, σ2 i ), ı = 1, 2, . . . , n.Find the moment-generating function of the random variable Y = X1 + · · · + Xn.
3. Suppose that X1, . . . , Xn are independent random variables such that Xi (i = 1, 2, . . . , n) is distributed with probability function P(Xi = xi) = p(xi) = e−λλxi xi! , xi = 0, 1, 2, . . . , n.Find the moment-generating function of the random variable Y = X1 + · · · + Xn.
2. Suppose that X1, . . . , Xn is a random sample from a normal population with meanμ and variance σ2. Find the moment-generating function of the sample mean ¯X = n i=1 Xi/n.
1. Suppose that X1, . . . , Xn are independent continuous random variables such that Xi (i = 1, 2, . . . , n) is gamma distributed with parameters γi and λ. Find the moment-generating function of the random variable Y = X1 + · · · + Xn.
5. Referring to Problem 9 of Section 6.2, find the moment-generating function of the random variable U = X − Y .
4. Suppose that X1, . . . , Xn are independent and identically distributed random variables with probability function for Xi given by P(Xi = xi) = p(xi) = e−λλxi xi! , xi = 0, 1, 2, . . . , n That is, the Xi’s constitute a random sample of n independent observations on X, where X has the
3. Let X1 and X2 be two independent random variables distributed as standard normal.Find the moment generating function of the random variable V = (2X1 + 3X2). Find the mean and variance of V .
2. Let (X1,X2,X3) be a random sample from a Poisson population with mean λ. Determine the moment-generating function of the random variable Y = 2X1 + 3X2 + X3 and use it to find the mean and the variance of the random variable Y .
1. Let X1 and X2 be two independent random variables distributed as standard normal.Find the moment-generating function of the random variable U = (X1 − X2). Find the mean and variance of U.
9. During rush hours, an engineer takes X minutes to reach to his/her office. This time includes (Y) the driving time from home to the parking lot and walking time from parking lot to the office. Thus, U = X − Y is the time that he/she has taken to find a parking spot. Suppose that the joint
8. The pair of random variables (X1,X2) has joint p.d.f. f(x1, x2) given by f(x1, x2) =2/π for (x1, x2) lying inside the semicircle bounded by the x1 axis and the curve x2 =1 − x21; that is, the sample space of (X1,X2) is S = {(x1, x2)|x21+ x22≤ 1, x2 ≥ 0}.Find the marginals of X1 and X2,
7. Referring to Example 6.2.7, find conditional probability functions p(x1|x2)and p(x2|x1), and evaluate E(X1|X2 = 0), V ar(X1|X2 = 0), E(X1|X2 = 1), V ar(X1|X2 = 1).
6. Referring to Example 6.2.6, find E(X1|X2 = x2), V ar(X1|X2 = x2), E(X2|X1 = x1), and V ar(X2|X1 = x1).
5. Referring to Example 6.2.6, find the marginal distributions of X1 and X2. What are the mean and variances of X1 and X2? Compute the correlation coefficient between X1 and X2.
4. If X1 and X2 have joint distribution f(x1, x2), show that Cov(X1,X2) = E(X1X2) −E(X1)E(X2). Hence show that if X1 and X2 are independent, then Cov(X1,X2) = 0.
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