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Probability And Statistical Inference 9th Edition Robert V. Hogg, Elliot Tanis, Dale Zimmerman - Solutions

A candy manufacturer selects mints at random from the production line and weighs them. For one week, the day shift weighed n1 = 194 mints and the night shift weighed n2 = 162 mints. The numbers of these mints that weighed at most 21 grams was y1 = 28 for the day shift and y2 = 11 for the night
An environmental survey contained a question asking what respondents thought was the major cause of air pollution in this country, giving the choices “automobiles,” “factories,” and “incinerators.” Two versions of the test, A and B, were used. Let pA and pB be the respective proportions
Let p equal the proportion of letters mailed in the Netherlands that are delivered the next day. Suppose that y = 142 out of a random sample of n = 200 letters were delivered the day after they were mailed.(a) Give a point estimate of p.(b) Use Equation 7.3-2 to find an approximate 90% confidence
Let p equal the proportion of Americans who favor the death penalty. If a random sample of n = 1234 Americans yielded y = 864 who favored the death penalty, find an approximate 95% confidence interval for p.
Let p equal the proportion of Americans who select jogging as one of their recreational activities. If 1497 out of a random sample of 5757 selected jogging, find an approximate 98% confidence interval for p.
A proportion, p, that many public opinion polls estimate is the number of Americans who would say yes to the question, “If something were to happen to the president of the United States, do you think that the vice president would be qualified to take over as president?” In one such random
Let p equal the proportion of college students who favor a new policy for alcohol consumption on campus. How large a sample is required to estimate p so that the maximum error of the estimate of p is 0.04 with 95% confidence when the size of the student body is (a) N = 1500? (b) N = 15,000? (c) N =
If Y1/n and Y2/n are the respective independent relative frequencies of success associated with the two binomial distributions b(n, p1) and b(n, p2), compute n such that the approximate probability that the random interval (Y1/n − Y2/n) ± 0.05 covers p1 − p2 is at least 0.80.
When placed in solutions of varying ionic strength, paramecia grow blisters in order to counteract the flow of water. The following 60 measurements in microns are blister lengths:(a) Construct an ordered stem-and-leaf diagram.(b) Give a point estimate of the median m = π0.50.(c) Find an
Let Y1 < Y2 < · · · < Y8 be the order statistics of eight independent observations from a continuous-type distribution with 70th percentile π0.7 = 27.3. (a) Determine P(y7 < 27.3). (b) Find P(Y5 < 27.3 < Y8).
For n = 12 year-2007 model sedans whose horsepower is between 290 and 390, the following measurements give the time in seconds for the car to go from 0 to 60 mph:(a) Find a 96.14% confidence interval for the median, m.(b) The interval (y1, y7) could serve as a confidence interval for Ï€0.3. Find
Let m denote the median weight of €œ80-pound€ bags of water softener pellets. Use the following random sample of n = 14 weights to find an approximate 95% confidence interval for m:(a) Find a 94.26% confidence interval for m.(b) The interval (y6, y12) could serve as a confidence interval for
A company manufactures mints that have a label weight of 20.4 grams. The company regularly samples from the production line and weighs the selected mints. During two mornings of production it sampled 81 mints, obtaining the following weights:(a) Construct an ordered stem-and-leaf display using
We would like to fit the quadratic curve y = Î²1 + Î²2x + Î²3x2 to a set of points (x1, y1), (x2, y2), . . . , (xn, yn) by the method of least squares. To do this, let(a) By setting the three first partial derivatives of h with respect to Î²1, Î²2, and Î²3 equal to 0, show that Î²1,
Show that the endpoints for a 100(1 ˆ’ Î³)% confidence interval for Î± are
Find 95% confidence intervals for Î±, Î², and Ïƒ2 for the predicted and earned grades data in Exercise 6.5-4.In Exercise 6.5-4
Using the cigarette data in Exercise 6.5-7, find 95% confidence intervals for Î±, Î², and Ïƒ2 under the usual assumptions.In Exercise 6.5-7
Obtain a two-sided 100(1 − γ)% prediction interval for the average of m future independent observations taken at the same X-value, x∗.
Using the ACT scores in Exercise 6.5-6, find 95% confidence intervals for Î±, Î², and Ïƒ2 under the usual assumptions.In Exercise 6.5-6
For the data given in Exercise 6.5-4, with the usual assumptions,In Exercise 6.5-4(a) Find a 95% confidence interval for Î¼(x) when x = 2, 3, and 4. (b) Find a 95% prediction interval for Y when x = 2, 3, and 4.
A computer center recorded the number of programs it maintained during each of 10 consecutive years. Year Number of Programs1 ........ 4302 ........ 4803 ........ 5654 ........ 7905 ........ 8856 ........ 9607 ........12008 ........13809 ........153010 ........1591(a) Calculate the least
By the method of least squares, fit the regression plane y = β1 + β2x1 + β3x2 to the following 12 observations of (x1, x2, y): (1, 1, 6), (0, 2, 3), (3, 0, 10), (–2, 0, –4), (–1, 2, 0), (0, 0, 1), (2, 1, 8), (–1, –1, –2), (0, –3, –3), (2, 1, 5), (1, 1, 1), (–1, 0, –2).
Consider the following 16 observed values, rounded to the nearest tenth, from the exponential distribution that was given in this section:(a) Take re-samples of size n = 16 from these observations about N = 200, times and compute s2 each time. Construct a histogram of these 200 bootstrapped values
Refer to the data in Example 7.5-2 and take resamples of size n = 27 exactly N = 500 times and compute the seventh order statistic, y7, each time.In Example 7.5-2(a) Construct a histogram of these N = 500 seventh order statistics.(b) Give a point estimate of Ï€0.25.(c) Find an 82% confidence
The following 54 pairs of data give, for Old Faithful geyser, the duration in minutes of an eruption and the time in minutes until the next eruption:(a) Calculate the correlation coefficient, and construct a scatterplot, of these data. (b) To estimate the distribution of the correlation
In a mechanical testing lab, Plexiglass strips are stretched to failure. Let X equal the change in length in mm before breaking. Assume that the distribution of X is N(Î¼, Ïƒ2). We shall test the null hypothesis H0: Î¼ = 5.70 against the alternative hypothesis
To test whether a golf ball of brand A can be hit a greater distance off the tee than a golf ball of brand B, each of 17 golfers hit a ball of each brand, 8 hitting ball A before ball B and 9 hitting ball B before ball A. The results in yards are as follows:Assume that the differences of the paired
A researcher claims that she can reduce the variance of N(μ,100) by a new manufacturing process. If S2 is the variance of a random sample of size n from this new distribution, she tests H0: σ2 = 100 against H1: σ2 < 100 by rejecting H0 if (n − 1)S2/100 ≤ χ1−α(n − 1) since (n −
Assume that the weight of cereal in a “12.6- ounce box” is N(μ, 0.22). The Food and Drug Association (FDA) allows only a small percentage of boxes to contain less than 12.6 ounces. We shall test the null hypothesis H0: μ = 13 against the alternative hypothesis H1: μ < 13. (a) Use a random
Let X equal the thickness of spearmint gum manufactured for vending machines. Assume that the distribution of X is N(Î¼, Ïƒ2). The target thickness is 7.5 hundredths of a n inch. We shall test the null hypothesis H0: Î¼ = 7.5 against a two-sided alternative
Let X equal the forced vital capacity (FVC) in liters for a female college student. (The FVC is the amount of air that a student can force out of her lungs.) Assume that the distribution of X is approximately N(Î¼, Ïƒ2). Suppose it is known that Î¼ = 3.4 liters.
A company that manufactures brackets for an automaker regularly selects brackets from the production line and performs a torque test. The goal is for mean torque to equal 125. Let X equal the torque and assume that X is N(Î¼, Ïƒ2). We shall use a sample of size n = 15 to test
Plants convert carbon dioxide (CO2) in the atmosphere, along with water and energy from sunlight, into the energy they need for growth and reproduction. Experiments were performed under normal atmospheric air conditions and in air with enriched CO2 concentrations to determine the effect on plant
Let X and Y denote the respective lengths of male and female green lynx spiders. Assume that the distributions of X and Y are N(μX, σ2x) and N(μY, σ2Y), respectively, and that σ2Y > σ2x. Thus, use the modification of Z to test the hypothesis H0: μX − μY = 0 against the alternative
An ecology laboratory studied tree dispersion patterns for the sugar maple, whose seeds are dispersed by the wind, and the American beech, whose seeds are dispersed by mammals. In a plot of area 50 m by 50 m, they measured distances between like trees, yielding the following distances in meters for
To measure air pollution in a home, let X and Y equal the amount of suspended particulate matter (in μg/m3) measured during a 24-hour period in a home in which there is no smoker and a home in which there is a smoker, respectively. We shall test the null hypothesis H0: σ2X/σ2Y = 1 against the
Let X and Y denote the weights in grams of male and female common gallinules, respectively. Assume that X is N(μX, σ2x) and Y is N(μY, σ2Y).(a) Given n = 16 observations of X and m = 13 observations of Y, define a test statistic and a critical region for testing the null hypothesis H0: μX =
Among the data collected for the World Health Organization air quality monitoring project is a measure of suspended particles, in μg/m3. Let X and Y equal the concentration of suspended particles in μg/m3 in the city centers (commercial districts), of Melbourne and Houston, respectively. Using n
Let X and Y equal the forces required to pull stud No. 3 and stud No. 4 out of a window that has been manufactured for an automobile. Assume that the distributions of X and Y are N(Î¼X, Ïƒ2x) and N(Î¼Y, Ïƒ2Y), respectively.(a) If m = n = 10 observations
Let X and Y denote the tarsus lengths of male and female grackles, respectively. Assume that X is N(μX, σ2x) and Y is N(μY, σ2Y). Given that n = 25, x = 33.80, s2 x = 4.88, m = 29, y = 31.66, and s2y = 5.81, test the null hypothesis H0: μX = μY against H1: μX > μY with α = 0.01.
Because of tourism in the state, it was proposed that public schools in Michigan begin after Labor Day. To determine whether support for this change was greater than 65%, a public poll was taken. Let p equal the proportion of Michigan adults who favor a post–Labor Day start. We shall test H0: p =
Let p equal the proportion of yellow candies in a package of mixed colors. It is claimed that p = 0.20.(a) Define a test statistic and critical region with a significance level of Î± = 0.05 for testing H0: p = 0.20 against a two-sided alternative hypothesis(b) To perform the test, each
For developing countries in Africa and the Americas, let p1 and p2 be the respective proportions of babies with a low birth weight (below 2500 grams). We shall test H0: p1 = p2 against the alternative hypothesis H1: p1 > p2. (a) Define a critical region that has an α = 0.05 significance level. (b)
Let p be the fraction of engineers who do not understand certain basic statistical concepts. Unfortunately, in the past, this number has been high, about p = 0.73. A new program to improve the knowledge of statistical methods has been implemented, and it is expected that under this program p would
A bowl contains two red balls, two white balls, and a fifth ball that is either red or white. Let p denote the probability of drawing a red ball from the bowl. We shall test the simple null hypothesis H0: p = 3/5 against the simple alternative hypothesis H1: p = 2/5. Draw four balls at random from
Let p denote the probability that, for a particular tennis player, the first serve is good. Since p = 0.40, this player decided to take lessons in order to increase p. When the lessons are completed, the hypothesis H0: p = 0.40 will be tested against H1: p > 0.40 on the basis of n = 25 trials. Let
It was claimed that 75% of all dentists recommend a certain brand of gum for their gum-chewing patients. A consumer group doubted this claim and decided to test H0: p = 0.75 against the alternative hypothesis H1: p < 0.75, where p is the proportion of dentists who recommend that brand of gum. A
Let p equal the proportion of drivers who use a seat belt in a state that does not have a mandatory seat belt law. It was claimed that p = 0.14. An advertising campaign was conducted to increase this proportion. Two months after the campaign, y = 104 out of a random sample of n = 590 drivers were
Let X and Y denote the heights of blue spruce trees, measured in centimeters, growing in two large fields. We shall compare these heights by measuring 12 trees selected at random from each of the fields. Take Î± ‰ˆ 0.05, and use the statistic W€”the sum of the ranks of the observations of
A charter bus line has 48-passenger and 38- passenger buses. Let m48 and m38 denote the median number of miles traveled per day by the respective buses. With Î± = 0.05, use the Wilcoxon statistic to test H0: m48 = m38 against the one-sided alternative H1: m48 > m38. Use the following data,
In Exercise 8.2-13, data are given that show the effect of a certain fertilizer on plant growth. The growths of the plants in mm over six weeks are repeated here, where Group A received fertilizer and Group B did not:We shall test the hypothesis that fertilizer enhanced the growth of the plants.(a)
Data were collected during a step-direction experiment in the biomechanics laboratory at Hope College. The goal of the study is to establish differences in stepping responses between healthy young and healthy older adults. In one part of the experiment, the subjects are told in what direction they
In Exercise 8.2-10, growth data are given for plants in normal air and for plants in CO2-enriched air. Those data are repeated here:In this exercise, we shall test the null hypothesis that the medians are equal, namely, H0: mX = mY, against the alternative hypothesis H1: mX (a) What is your
A course in economics was taught to two groups of students, one in a classroom situation and the other online. There were 24 students in each group. The students were first paired according to cumulative grade point averages and background in economics, and then assigned to the courses by a flip of
The outcomes on n = 10 simulations of a Cauchy random variable were −1.9415, 0.5901, −5.9848, −0.0790, −0.7757, −1.0962, 9.3820, −74.0216, −3.0678, and 3.8545. For the Cauchy distribution, the mean does not exist, but for this one, the median is believed to equal zero. Use the
Let m equal the median of the posttest grip strengths in the right arms of male freshmen in a study of health dynamics. We shall use observations on n = 15 such students to test the null hypothesis H0: m = 50 against the alternative hypothesis H1: m > 50.(a) Using the Wilcoxon statistic, define
A pharmaceutical company is interested in testing the effect of humidity on the weight of pills that are sold in aluminum packaging. Let X and Y denote the respective weights of pills and their packaging (in grams), when the packaging is good and when it is defective, after the pill has spent 1
Let X1, X2, ... , X8 be a random sample of size n = 8 from a Poisson distribution with mean λ. Reject the simple null hypothesis H0: λ = 0.5, and accept H1: λ > 0.5, if the observed sum 8i=1 xi ≥ 8. (a) Compute the significance level α of the test. (b) Find the power function K(λ) of the
Let X1, X2, X3 be a random sample of size n = 3 from an exponential distribution with mean θ > 0. Reject the simple null hypothesis H0: θ = 2, and accept the composite alternative hypothesis H1: θ < 2, if the observed sum 3i=1 xi ≤ 2. (a) What is the power function K(θ), written as an
Let X equal the number of milliliters of a liquid in a bottle that has a label volume of 350 ml. Assume that the distribution of X is N(Î¼, 4). To test the null hypothesis H0: Î¼ = 355 against the alternative hypothesis H1: Î¼ (a) Find the power function K(Î¼) for this test.(b) What is the
Let X be N(μ,100). To test H0: μ = 80 against H1: μ > 80, let the critical region be defined by C = {(x1, x2, ... , x25) : x ≥ 83}, where x is the sample mean of a random sample of size n = 25 from this distribution. (a) What is the power function K(μ) for this test? (b) What is the
Let X equal the butterfat production (in pounds) of a Holstein cow during the 305-day milking period following the birth of a calf. Assume that the distribution of X is N(Î¼,1402). To test the null hypothesis H0: Î¼ = 715 against the alternative hypothesis H1: Î¼ (a) Find the power function
Let X have a Bernoulli distribution with pmfWe would like to test the null hypothesis H0: p ‰¤ 0.4 against the alternative hypothesis H1: p > 0.4. For the test statistic, use is a random sample of size n from this Bernoulli distribution. Let the critical region be of the form C = {y: y
Let X1, X2, . . . , Xn be a random sample from N(0, Ïƒ2).(a) Show that C = {(X1, X2, . . . , Xn) :Is a best critical region for testing H0: Ïƒ2 = 4 against H1: Ïƒ2 = 16. (b) If n = 15, find the value of c so that Î± = 0.05. (c) If n = 15 and c is the
Let X1, X2, . . . , Xn be a random sample of Bernoulli trials b(1, p).(a) Show that a best critical region for testing H0: p = 0.9 against H1: p = 0.8 can be based on the statisticc) What is the approximate value of Î² = P[ Y > n(0.85); p = 0.8 ] for the test given in part (b)? (d) Is
Let X1, X2, . . . , Xn be a random sample from the normal distribution N(Î¼, 9). To test the hypothesis H0: Î¼ = 80 against H1: Î¼ ‰ 80, consider the following three critical regions: C1 = {: ‰¥ C1}, C2 = {: ‰¤ C2}, and C3 = {: | ˆ’ 80| ‰¥ C3}.(a) If n = 16, find the
Consider a random sample X1, X2, . . . , Xn from a distribution with pdf f(x; θ) = θ(1 − x)θ−1, 0 < x < 1, where 0 < θ. Find the form of the uniformly most powerful test of H0: θ = 1 against H1: θ > 1.
Referring back to Exercise 6.4-19, find the likelihood ratio test of H0: γ = 1, μ unspecified, against all alternatives.
Assume that the weight X in ounces of a “10- ounce” box of cornflakes is N(μ, 0.03). Let X1, X2, . . . , Xn be a random sample from this distribution. (a) To test the hypothesis H0: μ ≥ 10.35 against the alternative hypothesis H1: μ < 10.35, what is the critical region of size α = 0.05
Let X1, X2, . . . , Xn be a random sample of size n from the normal distribution N(μ, σ02), where σ02 is known but μ is unknown.(a) Find the likelihood ratio test for H0: μ = μ0 against H1: μ ≠ μ0. Show that this critical region for a test with significance level α is given by |− μ0|
To test H0: μ = 335 against H1: μ < 335 under normal assumptions, a random sample of size 17 yielded = 324.8 and s = 40. Is H0 accepted at an α = 0.10 significance level?
Let X1, X2, ... , Xn be a random sample from an exponential distribution with mean Î¸. Show that the likelihood ratio test of H0: Î¸ = Î¸0 against H1: Î¸ ‰ Î¸0 has a critical region of the formHow would you modify this test so
A biologist is studying the life cycle of the avian schistosome that causes swimmer€™s itch. His study uses Menganser ducks for the adult parasites and aquatic snails as intermediate hosts for the larval stages. The life history is cyclic. (For more information, see swimmersitch.org/.) As a
A particular brand of candy-coated chocolate comes in five different colors that we shall denote as A1 = {brown}, A2 = {yellow}, A3 = {orange}, A4 = {green}, and A5 = {coffee}. Let pi equal the probability that the color of a piece of candy selected at random belongs to Ai, i = 1, 2,..., 5. Test
In a biology laboratory, students use corn to test the Mendelian theory of inheritance. The theory claims that frequencies of the four categories “smooth and yellow,” “wrinkled and yellow,” “smooth and purple,” and “wrinkled and purple” will occur in the ratio 9:3:3:1. If a student
It has been claimed that, for a penny minted in 1999 or earlier, the probability of observing heads upon spinning the penny is p = 0.30. Three students got together, and they would each spin a penny and record the number X of heads out of the three spins. They repeated this experiment n = 200
For determining the half-lives of radioactive isotopes, it is important to know what the background radiation is for a given detector over a certain period. A Î³-ray detection experiment over 300 one-second intervals yielded the following data:Do these look like observations of a Poisson
A random sample of 50 women who were tested for cholesterol was classified according to age and cholesterol level and grouped into the following contingency table.
In a psychology experiment, 140 students were divided into majors emphasizing left-hemisphere brain skills (e.g., philosophy, physics, and mathematics) and majors emphasizing right-hemisphere skills (e.g., art, music, theater, and dance). They were also classified into one of three groups on the
Suppose that a third group of nurses was observed along with groups I and II of Exercise 9.2-1, resulting in the respective frequencies 130, 75, 136, 33, 61, and 65. Test H0: pi1 = pi2 = pi3, i = 1, 2, . . . , 6, at the α = 0.025 significance level.
Suppose that a third class (W) of 15 students was observed along with classes U and V of Exercise 9.2-3, resulting in scores ofAgain, use a chi-square test with Î± = 0.05 to test the equality of the three distributions by dividing the combined sample into three equal parts.
A random survey of 100 students asked each student to select the most preferred form of recreational activity from five choices. Following are the results of the survey:Test whether the choice is independent of the gender of the respondent. Approximate the p-value of the test. Would we reject the
A student who uses a certain college€™s recreational facilities was interested in whether there is a difference between the facilities used by men and those used by women. Use Î± = 0.05 and the following data to test the null hypothesis that facility and gender are
From the box-and-whisker diagrams in Figure 9.3-1, it looks like the means of X1 and X2 could be equal and also that the means of X3, X4, and X5 could be equal but different from the first two.(a) Using the data in Example 9.3-2, as well as a t test and an F test, test H0: Î¼1 =
A particular process puts a coating on a piece of glass so that it is sensitive to touch. Randomly throughout the day, pieces of glass are selected from the production line and the resistance is measured at 12 different locations on the glass. On each of three different days, December 6, December
Ledolter and Hogg (see References) report the comparison of three workers with different experience who manufacture brake wheels for a magnetic brake. Worker A has four years of experience, worker B has seven years, and worker C has one year. The company is concerned about the product€™s
Let Î¼i be the average yield in bushels per acre of variety i of corn, i = 1, 2, 3, 4. In order to test the hypothesis H0: Î¼1 = Î¼2 = Î¼3 = Î¼4 at the 5% significance level, four test plots for each of the four varieties of corn are planted. Determine whether we accept or reject H0 if
Ledolter and Hogg report that a civil engineer wishes to compare the strengths of three different types of beams, one (A) made of steel and two (B and C) made of different and more expensive alloys. A certain deflection (in units of 0.001 inch) was measured for each beam when submitted to a given
Let X1, X2, X3, X4 equal the cholesterol level of a woman under the age of 50, a man under 50, a woman 50 or older, and a man 50 or older, respectively. Assume that the distribution of Xi is N(Î¼i, Ïƒ2), i = 1, 2, 3, 4. We shall test the null hypothesis H0: Î¼1 = Î¼2 = Î¼3 = Î¼4, using
Different sizes of nails are packaged in €œ1-pound€ boxes. Let Xi equal the weight of a box with nail size C, i = 1, 2, 3, 4, 5, where 4C, 8C, 12C, 16C, and 20C are the sizes of the sinkers from smallest to largest. Assume that the distribution of Xi is N(Î¼i, Ïƒ2). To test the null
With a = 3 and b = 4, find Î¼, Î±i, and Î²j if Î¼ij, i = 1, 2, 3 and j = 1, 2, 3, 4, are given byIn an €œadditive€ model such as this one, one row (column) can be determined by adding a constant value to each of the elements of another row (column).
Show that the cross-product terms formed from (i· ˆ’ ··), (·j ˆ’ ··), and (ij ˆ’ i· ˆ’ ·j + X··) sum to zero, i = 1, 2, . . . a and j = 1, 2, . . . , b. For example, writeAnd sum each term
With a = 3 and b = 4, find Î¼, Î±i, Î²j, and Î³ij if Î¼ij, i = 1, 2, 3 and j = 1, 2, 3, 4, are given byNote the difference between the layout here and that in Exercise 9.4-2. Does the interaction help explain the difference?
There is another way of looking at Exercise 9.3-6, namely, as a two-factor analysis-of-variance problem with the levels of gender being female and male, the levels of age being less than 50 and at least 50, and the measurement for each subject being their cholesterol level. The data would then be
9 . 5 - 4 . Ledolter and Hogg (see References) note that percent yields from a certain chemical reaction for changing temperature (factor A), reaction time (factor B), and concentration (factor C) are x1 = 79.7, x2 = 74.3, x3 = 76.7, x4 = 70.0, x5 = 84.0, x6 = 81.3, x7 = 87.3, and x8 = 73.7, in
In a college health fitness program, let X equal the weight in kilograms of a female freshman at the beginning of the program and let Y equal her change in weight during the semester. We shall use the following data for n = 16 observations of (x, y) to test the null hypothesis H0: Ï = 0 against
In Exercise 6.5-5, data are given for horsepower, the time it takes a car to go from 0 to 60, and the weight in pounds of a car, for 14 cars. Those data are repeated here:(a) Let Ï be the correlation coefficient of horsepower and weight. Test H0: Ï = 0 against H1: Ï
For the data given in Exercise 6.5-4, use a t test to test H0: β = 0 against H1: β > 0 at the α = 0.025 significance level.In Exercise 6.5-4Ledolter and Hogg (see References) note that percent yields from a certain chemical reaction for changing temperature (factor A), reaction time (factor
In bowling, it is often possible to score well in the first game and then bowl poorly in the second game, or vice versa. The following six pairs of numbers give the scores of the first and second games bowled by the same person on six consecutive Tuesday evenings:Assume a bivariate normal
By squaring the binomial expression [(Yi ˆ’) ˆ’ (SxY/s2x)(xi ˆ’ )], show that

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