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Essential Statistics For The Behavioral Sciences 2nd Edition Gregory J Privitera - Solutions

7. It is believed that drinking has some bad effects on the human reproduction system.To study this, some evaluations of placenta tissue of 16 randomly selected drinking mothers were made that yielded the following values (recorded to the nearest whole number):18 17 22 21 15 21 22 22 14 20 14 16 13
6. The weights of a random sample of 49 university male first-year students yielded a mean of 165 pounds and a standard deviation of 6.5 pounds. Determine a 90%confidence interval for the mean weight of all university male first-year students.
5. The following data give the drying time (in hours) for 10 randomly selected concrete slabs:9.06 9.17 9.11 8.16 9.10 9.98 8.89 9.02 9.32 8.12 Assuming that drying times are normally distributed, determine a 95% confidence interval for the mean drying time for the slabs.
4. A study was undertaken to see if the length of slide pins used in the front disc brake assembly met with specifications. To this end, measurements of the lengths of 16 slide pins, selected at random, were made. The average value of 16 lengths was 3.15, with a sample standard deviation of 0.2.
3. An insurance company is interested in determining the average postoperative length of stay (in days) in hospitals for all patients who have bypass surgery. The following data give length of stay of 50 randomly selected patients who had bypass surgery:6 10 10 9 9 12 7 12 7 8 10 7 8 8 10 12 10 7 7
2. Suppose that in Problem 1 only 25 of 36 randomly selected family caregivers responded, so we have the following data:55 53 47 47 49 43 47 40 48 41 44 51 48 43 50 49 47 42 42 47 47 49 46 46 43 Assuming that these data come from a population that has a normal distribution,(a) Determine a 99%
1. The following data give the ages of 36 randomly selected family caregivers of older parents in the United States:55 53 47 47 49 43 47 40 48 41 44 51 48 43 50 49 47 42 42 47 47 49 46 46 43 41 45 51 44 48 43 50 53 44 49 53 Assuming normality,(a) Determine a 95% confidence interval for the mean
10. Referring to Problem 9,(a) Show that U = (n + 1)ˆθ/n is unbiased for θ.(b) Find the variance of U.(c) Verify that 2¯X is unbiased for θ. Find the variance of 2¯X .(d) Determine the ratio V ar(U)/V ar(2¯X ). Which of the unbiased estimators would you prefer?
9. Suppose (X1, . . . , Xn) is a random sample of n independent observations from a population having p.d.f.f(x) =1θ , 0 ≤ x ≤ θ0, otherwise Find the MLE of θ. If ˆθ is the MLE of θ, then show that ˆθ is not unbiased for θ.
8. Events occur in time in such a way that the time interval between two successive events is a random variable t having the p.d.f. θe−θt, θ>0. Suppose that observations are made of the n successive time intervals for n + 1 events, yielding(t1, . . . , tn). Assuming these time intervals to be
7. If (X1, . . . , Xn) is a random sample of size n from a population having a Poisson distribution with unknown parameter λ, find the MLE for λ.
6. Suppose that a random sample of size n is taken from a gamma distribution with parameters γ and λ. Find the method of moments estimators of γ and λ.
5. The following data give the pull strength of 20 randomly selected solder joints on a circuit board. Assume that the pull strengths are normally distributed with meanμ and variance σ2.12 11 12 9 8 11 11 11 8 9 10 9 8 11 10 8 9 9 11 10(a) Determine the maximum likelihood estimate of the
4. Suppose that S2 1 and S2 2 are sample variances of two samples of n1 and n2 independent observations, respectively, from a population with mean μ and variance σ2. Determine an unbiased estimator of σ2 as a combination of S2 1 and S2 2 .8.3 Interval Estimators for the Mean μ of a Normal
3. Let X1,X2, and X3 be independent random variables with mean μ and varianceσ2. Suppose that ˆμ1 and ˆμ2 are two estimators of μ, where ˆμ1 = 2X1 − 2X2 + X3 and ˆμ2 = 2X1 − 3X2 + 2X3.(a) Show that both estimators are unbiased for μ.(b) Find the variance of each estimator and
2. The lengths of a random sample of 20 rods produced the following data:12.2 9.5 13.2 13.9 9.5 9.5 11.9 9.2 11.0 10.4 9.9 12.8 10.5 11.9 12.3 10.0 8.7 6.2 10.0 11.2 Determine the method of moments estimate of μ and σ2, assuming that the rod lengths are normally distributed with mean μ and
1. Find a method of moments estimator of the mean of a Bernoulli distribution with parameter p.
20. Seven engineers in a manufacturing company are working on a project. Let random variables T1, . . . , T7 denote the time (in hours) needed by the engineers to finish the project. Suppose that T1, . . . , T7 are independently and identically distributed by the uniform distribution over an
19. Repeat Problems 17 and 18, by supposing that the lifetimes of the three components are independently and identically distributed as Weibull with α = 2, β = 0.5.
18. Suppose that in Problem 17, the components are in parallel, so that the system will fail only when all the components fail. Find the p.d.f. of T, and then find the probability P(T > 15).
17. A mechanical system has three components in series, so the system will fail when at least one of the components fails. The random variables X1,X2, and X3 represent the lifetime of these components. Suppose that the Xi(i = 1, 2, 3) are independently and identically distributed as exponential
16. In Problem 15, using MINITAB, R, or JMP, find the probabilities: (a) P(S2 > 100),(b) P(S2 > 130), (c) P(S2 > 140).
15. Suppose that the total cholesterol levels of the US male population between 50 and 70 years of age are normally distributed with mean 170 mg/dL and standard deviation 12 mg/dL. Let X1, . . . , X21 be the cholesterol levels of a random sample of 21 US males between the ages of 50 and 70 years.
14. Refer to Problem 13. Using MINITAB, R, or JMP, find the probabilities: (a)P(S2x/S2 y > 3.5), (b) P(2.0 < S2x/S2 y < 3.5).
13. Let X1, . . . , X16 and Y1, . . . , Y13 be two independent random samples from two normal populations with equal variances. Show that the p.d.f. of S2x/S2 y is distributed as Snedecor’s F15,12.
12. Find, using MINITAB/R/JMP, the value of x such that(a) P(5 ≤ χ2 16 ≤ x) = 0.95(b) P(10 ≤ χ2 20 ≤ x) = 0.90(c) P(0.5 ≤ F20,24 ≤ x) = 0.90(d) P(0.4 ≤ F12,15 ≤ x) = 0.85
11. Suppose that X(1), . . . , X(n) are the order statistics of a sample from a population having the rectangular distribution with p.d.f.f(x) =⎧⎪⎨⎪⎩0, x≤ 0 1θ , 0 < x ≤ θ0, x> θwhere θ is an unknown parameter. Show that for 0 < γ < 1, PX(n) ≤ θ ≤X(n)n√1 − γ= γ(Note:
10. In Problem 9, show that for 1 ≤ k ≤ n, the mean and variance of F(x(k)) are respectively kn + 1 and k(n − k + 1)(n + 1)2(n + 2)
9. If X(1), . . . , X(n) are the order statistics of a sample of size n from a population having a continuous c.d.f. F(x) and p.d.f. f(x), show that F(x(n)) has mean n/(n + 1) and variance n/[(n + 1)2(n + 2)].
8. A sample of n observations is taken at random from a population with p.d.f.f(x) =e−x, x≥ 0 0, x< 0 Find the p.d.f. of the smallest observation. What are its mean and variance? What is its c.d.f.?
7. Suppose that the diameter of ball bearings used in heavy equipment are manufactured in a certain plant and are normally distributed with mean 1.20 cm and a standard deviation 0.05 cm. What is the probability that the average diameter of a sample of size 25 will be(a) Between 1.18 and 1.22 cm?(b)
6. Suppose that n = 2m + 1 observations are taken at random from a population with probability density function f(x) =⎧⎪⎪⎨⎪⎪⎩0, x≤ a 1b−a, a b Find the distribution of the median of the observations and find its mean and variance.What is the probability that the median will exceed
5. Suppose that F(x) is the fraction of bricks in a very large lot having crushing strengths of x psi or less. If 100 such bricks are drawn at random from the lot:(a) What is the probability that the crushing strengths of all 100 bricks exceed x psi?Review Practice Problems 287(b) What is the
4. It is believed that the median annual starting salary of a fresh engineering graduate is$40,000. If we take a random sample of 100 recent engineering graduates and record their starting salary, then(a) Find the sampling distribution of ˆp, the proportion of fresh engineering graduates who
3. A manufacturer of car batteries finds that 80% of its batteries last more than five years without any maintenance. Suppose that the manufacturer took a random sample of 500 persons from those who bought those batteries and recorded the lifetimes of their batteries.(a) Find the sampling
2. Suppose that in a certain country, the ages of women at the time of death are distributed with mean 70 years and standard deviation 4 years. Find the approximate probability that the average age of a randomly selected group of 36 women will be (a)more than 75 years, (b) less than 70 years, (c)
1. The times taken by all students of a large university to complete a calculus test are distributed having a mean of 120 minutes and a standard deviation of 10 minutes.Calculate the approximate probability that the average time taken to complete their test by a random sample of 36 students will be
8. Consider a system of n identical components operating independently. Suppose the lifetime, in months, is exponentially distributed with mean 1/λ. These components are installed in series, so that the system fails as soon as the first component fails.Find the probability density function of the
7. In Problem 5, assume that n = 21.(a) Find the probability density function of the median time taken by the manager to drive from one plant to another.(b) Find the expected value of X(21).(c) Find the expected value of X(11), the median.
6. The lifetime, in years, X1,X2, · · · ,Xn of n randomly selected power steering pumps manufactured by a subsidiary of a car company is exponentially distributed with mean 1/λ. Find the probability density function of X(1) =Min(X1,X2, · · · ,Xn), and find its mean and variance.286 7
5. The time, in minutes, taken by a manager of a company to drive from one plant to another is uniformly distributed over an interval [15, 30]. Let X1,X2, . . . , Xn denote her driving times on n randomly selected days, and let X(n) =Max(X1,X2, · · · ,Xn).Determine(a) The probability density
4. Suppose F(x) is the fraction of objects in a very large lot having weights less than or equal to x pounds. If 10 objects are drawn at random from the lot:(a) What is the probability that the heaviest of 10 objects chosen at random without replacement will have a weight less than or equal to u
3. Assume that the cumulative distribution function of breaking strengths (in pounds)of links used in making a certain type of chain is given by F(x) =1 − e−λx, x> 0 0, x≤ 0 where λ is a positive constant. What is the probability that a 100-link chain made from these links would have a
2. If ten points are picked independently and at random on the interval (0, 1):(a) What is the probability that the point nearest 1 (i.e., the largest of the 10 numbers selected) will lie between 0.9 and 1.0?(b) The probability is 1/2 that the point nearest 0 will exceed what number?
1. A continuous random variable, say X, has the uniform distribution function on (0, 1)so that the p.d.f. of X is given by f(x) =⎧⎨⎩0, x≤ 0 1, 0 < x ≤ 1 0, x>1 If X(1),X(2), . . . , X(n) are the order statistics of n independent observations all having this distribution function, give the
7. Find the value of x such that (a) P(3.247 < χ2 10 < x) = 0.95, (b) P(8.260 < χ2 20
6. Suppose that the random variable T has the Student t-distribution with 24 degrees of freedom. Find the value of t such that (a) P(−1.318 < T < t) = 0.80, (b)P(−1.711 < T < t) = 0.85, (c) P(−2.064 < T < t) = 0.875.
5. Use MINITAB, R, or JMP to do the Problems 1, 2, 3, and 4 above.
4. Use Table A.7 to find the following values: (a) F10,12,0.95, (b) F8,10,0.975, (c) F15,20,0.95,(d) F20,15,0.99. Hint: Use the formula Fm,n,1−α = 1/Fn,m,α.
3. Use Table A.7 to find the following values of upper percent points of various F-distributions: (a) F6,8,0.05, (b) F8,10,0.01, (c) F6,10,0.05, (d) F10,11,0.025
2. Use Table A.6 to find the following values of upper percent points of various tm-distributions: (a) t18,0.025, (b) t20,0.05, (c) t15,0.01, (d) t10,0.10, (d) t12,0.005.
1. If X is a chi-square random variable with 15 degrees of freedom, find the value of x such that (a) P(X ≥ x) = 0.05, (b) P(X ≥ x) = 0.975, (c) P(X ≤ x) = 0.025, (d)P(X ≥ x) = 0.95, (e) P(X ≤ x) = 0.05
8. The amount of beverage dispensed by a bottling machine is normally distributed with mean of 12 oz and a standard deviation of 1 oz. A random sample of n bottles is selected, and a sample average ¯X is calculated. Determine the following probabilities:(a) P(|¯X − 12| ≤ 0.25) for sample
7. In 1995, the median price of a PC was $1200. Suppose that a random sample of 100 persons who bought their PCs during that year recorded the amount spent (by each of them) on his/her PC. State the approximate sampling distribution of ˆp, the proportion of persons who spent more than $1200 on a
6. Let X be a random variable distributed as binomial B(n, p). State the approximate sampling distribution of the sample proportion ˆp when (a) n = 40, p = 0.4;(b) n = 50, p = 0.2; (c) n = 80, p = 0.1.
5. Suppose that the amount of a weekly grocery bill of all households in a metropolitan area is distributed with a mean of $140 and a standard deviation of $35. Let ¯X be the average amount of grocery bill of a random sample of 49 households selected from this metropolitan area. Find the
4. The weight of all cars traveling on interstate highway I-84 is normally distributed with a mean of 3000 lb and standard deviation of 100 lb. Let ¯X be the mean weight of a random sample of 16 cars traveling on I-84. Calculate the probability that ¯X falls between 2960 and 3040 lb.
3. Suppose that random samples are drawn from an infinite population. How does the standard error of ¯X change if the sample size is (i) increased from 36 to 64, (ii)increased from 100 to 400, (iii) increased from 81 to 324, (iv) increased from 256 to 576?
2. Suppose that random samples (without replacement) of size 5 are repeatedly drawn from a finite population of size N = 50. Suppose that the mean and the standard deviation of the population are 18 and 5, respectively. Find the mean and the standard deviation of the sampling distribution of ¯X.
1. Suppose that random samples of size 36 are repeatedly drawn from a population with mean 28 and standard deviation 9. Describe the sampling distribution of ¯X .
6. Refer to Problem 5. Suppose that the instructor of the class decided to give 10 extra points to every student. Find the mean and variance of the new data and comment on your result.
5. The following data give the scores on a midterm test of 20 randomly selected students.29 26 40 27 35 39 37 40 37 34 36 28 26 33 37 25 27 33 26 29 Find the mean and standard deviation for these data.
4. A manufacturing company has developed a new device for the army, obtaining a defense contract to supply 25,000 pieces of this device to the army. In order to meet the contractual obligations, the department of human resources wants to estimate the number of workers that the company would need to
3. Refer to Problem 2. Let T denote the total expenses for entertainment of all the students. Estimate the mean and variance of T.
2. The monthly entertainment expenses to the nearest dollar of 10 college students randomly selected from a university with 10,000 students are as follows:48 46 33 40 29 38 37 37 40 48 Determine the mean and standard deviation of these data.
1. Define the appropriate population from which the following samples have been drawn:(a) Fifty employees from a manufacturing company are asked if they would like the company to have some training program for all employees.(b) A quality control engineer of a semiconductor company randomly selects
Skewness and the assumptions for t tests. Rietveld and van Hout (2015) explained that skewness is an important feature of the data as it relates to “assumptions of t tests” (p. 158). Which assumption does skewness relate to? Explain.
Sociodemographic differences in lung cancer worry. Hahn (2017) evaluated sociodemographic differences in how people worry about lung cancer. Some of the differences observed across demographics of interest were between males and females [t(45) = 0.69; higher mean worry among men], smokers and
Assumptions for the two-independent-sample t test. Bakker and Wicherts (2014) identified that researchers will often exclude outliers in data sets that are nonnormal prior to conducting a two-independent-sample t test. While their findings showed that the practice of removing outliers was not
Distress and coping among high school students. In a comparison of coping strategies used among distressed and nondistressed high school students, Lin and Yusoff (2013) identified the following 95% CI for the difference in the use of self-blame as a coping strategy: −0.76 (−1.08 to−0.45). If
Athletics and academic achievement among elementary school children. Dyke (2014)showed that students who participated in school sports had higher standardized test scores in both reading and math than those who did not, in a sample of 1,605 fourth- and fifth-grade boys and girls. What statistical
Identifying tests for two samples. Özdemir (2013) stated, “when trying to detect and describe differences between two independent groups, by far the most common strategy is to use the arithmetic mean as a measure of location and Student’s t test as a method” (p. 322). What is the name of the
Power and degrees of freedom. In an explanation regarding the use of t tests, Zimmerman(2012) implied that power is greater for a test associated with n − 1 degrees of freedom compared to a test associated with N − 2 degrees of freedom—assuming that the total sample sizes are equal (i.e., n =
Using Cohen’sd, state whether a 3-point treatment effect (M1 − M2 = 3) is small, medium, or large for a two-independent-sample t test, given the following values for the pooled sample variance: 1. 2. 3. 4. S S S 2 P=9 2 P=36 2 Sp=576 S 2 P-144
A researcher reports a value of .28 for the effect size estimate. State the size of this effect as small, medium, or large, assuming the value is for (a) Cohen’sd, (b) eta-squared, and (c) omega-squared.
A researcher records the number of words recalled by students presented with a list of words for one minute. In one group, students were presented with the list of words in color; in a second group, the same words were presented in black and white. An equal number of students were in each group.
A social psychologist records the number of outbursts among students at two schools. Assuming the same number of students were observed at each school, what is the sample size at each school, the decision (to retain or reject the null hypothesis), and the effect size for this study, based on the
Individual actions can play a large role in the overall health of our planet. A researcher interested in evaluating environmentally friendly behaviors evaluated how often people recycle (per month)based on whether they have an overall optimistic or an overall pessimistic attitude toward ecofriendly
A study evaluating the effects of parenting style (authoritative, permissive) on child well-being observed 20 children (10 from parents who use an authoritative parenting style and 10 from parents who use a permissive parenting style). Children between the ages of 12 and 14 completed a standard
While researching lifestyle changes to improve happiness, you come across a research article reporting that pet owners self-reported greater overall happiness compared to those who do not own a pet, t(50) = 2.993, p < .05. Based on the information provided, answer the following questions:1. Does
A researcher reports the following confidence interval to determine whether or not there is a difference between two groups (i.e., whether or not the difference is larger than 0): 95% CI 2.0 [0.2 to 3.8].1. What would the decision have likely been if the researcher tested this hypothesis with
A researcher reports the following confidence interval for a comparison of the difference between two groups: 95% CI −12.0 [−30.0 to 6.0]. If there was no difference between groups, then a difference of 0 was expected.1. What would the decision have likely been if the researcher tested this
Will each of the following increase, decrease, or have no effect on the value of the test statistic for a two-independent-sample t test?1. The total sample size is increased.2. The level of significance is reduced from .05 to .01.3. The pooled sample variance is doubled.
State the critical values for a two-independent-sample t test given the following conditions:1. Two-tailed test, α = .01, total df = 26 2. One-tailed test, lower-tail critical, α = .01, df = 15 for each group 3. Two-tailed test, α = .05, n = 12 in each group 4. One-tailed test, upper-tail
Will each of the following increase, decrease, or have no effect on the critical value of a twoindependent-sample t test?1. The sample size is increased.2. The level of significance is reduced from .05 to .01.3. The pooled sample variance is doubled.
State the total degrees of freedom for the following t tests:1. n = 12 for a one-sample t test 2. Critical value = 1.645 for a one-tailed test, α = .05 3. df1 = 12, n2 = 19 for a two-independent-sample t test 4. Critical value = 63.657 for a two-tailed test, α = .01
State the degrees of freedom for the t test for each of the following situations in which two groups are observed.1. A study in which 12 participants are assigned to one group and 15 participants are assigned to a second group 2. A study in which 30 participants are selected, then half are observed
In the following studies, state whether you would use a one-sample t test or a two-independentsample t test.1. A study testing whether night-shift workers sleep the recommended 8 hours per day 2. A study measuring differences in attitudes about morality among men and women 3. An experiment
Using the same example as in Question 3, what is the proportion of variance using omega-squared?
The value of the test statistic for a two-independent-sample t test is 2.400. Using eta-squared, what is the proportion of variance when the degrees of freedom are 30?
If the difference between two means is 4, then what will the estimated Cohen’s d value be with a pooled sample standard deviation of each of the following?1. 4 2. 8 3. 1 4. 40
What is the denominator for computing an estimated Cohen’s d for a two-independent-sample t test?
Name two measures of proportion of variance. Which measure is the least conservative?
Name three measures used to estimate effect size for the two-independent-sample t test.
What is the calculation of the degrees of freedom for the two-independent-sample t test?
Explain how overlap in scores between two groups can help to identify whether a difference observed is likely to be significant.
Of the two ways to select two samples described in this chapter, which type of sampling is commonly used in experiments?
When an independent sample is selected, are the same or different participants observed in each group?
What is the between-subjects design?
Using the same example as in Question 3, what is the researcher’s decision if she reports the following values for the test statistic?1. 2.558 2. 1.602 3. 2.042 4. −2.500

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