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InFigure 2you may find a scatter plot with 10 points and 3 lines. Three of the points are marked by the letters a, b, and c. The lines are marked by the numbers 1, 2, and 3. The following 2 questions refer to this scatter plot.

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Fit a regression model to the relation between "curb.weight" as a response and "engine.size" as an explanatory variable. Both variables belong to the data set "cars.csv". The following 3 refer to the fitted model.

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The estimated value of the slope is equal to: (Give an answer of the form x.xxx with 3 significance digits)

Answer:

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Test the null hypothesis the intercept is equal to 0. This null hypothesis is:

Select one:

a.Not rejected with a significance level of 5%.

b.Rejected with a significance level of 5%, but not with a significance level of 1%.

c.Rejected with a significance level of 1%, but not with a significance level of 0.1%.

d.Rejected with a significance level of 0.1%

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Use the function "confint" in order to construct a confidence interval to the slope of the regression line. The value 0:

Select one:

a.Belongs to the 95% confidence interval.

b.Does not belongs to the 95% confidence interval.

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The next 3 questions refer to the data given in following table:Dataxy10.5-3.421.5-1.632.5-1.543.50.151.5-1.862.50.97-0.5-4.181.5-1.39-0.5-1.2101.5-2.9

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If y is the response and x is the explanatory variable then the slope of the regression line is equal to: (Give an answer of the form x.xxx with 3 significance digits)

Answer:

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If x is the response and y is the explanatory variable then the slope of the regression line is equal to: (Give an answer of the form x.xxx with 3 significance digits)

Answer:

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Consider a response and an explanatory variable. Assume that the variance of the explanatory variable is positive. If the covariance between the two variables is negative then:

Select one:

a.The regression line is decreasing

b.The regression line is increasing

c.The regression line may be increasing or decreasing.

d.The regression line has a zero slope.

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Consider a response and an explanatory variable. Assume that the variance of the explanatory variable is positive. If the regression line is increasing then covariance between x and y:

Select one:

a.is positive.

b.is negative.

c.may be positive or negative.

d.is equal to 0.

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A rectangular body is a "Golden Rectangle" if the ratio between the length and the width is equal to the golden ratio 1.618. Golden Rectangles are considered to be esthetic and are found, for example, in the structure of ancient Greek temples. In the quizzes of Unit 1 the possibility of testing that cars are square on the basis of the information in the data set "cars.csv" was explored. In this exercise we want to test the null hypothesis that the body of a car is a Golden Rectangle. Specifically, you are asked to apply the function "t.test" to the ratio between the length and the width of cars. (Hint: Use the code "r <- cars$length/cars$width" in order to produce "r" and apply the function to "r".)

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The null hypothesis in the application of the function "t.test" to the variable "r" corresponds to the statement:

Select one:

a.The ratio between the length of the car and the width of the car is equal to 1.618.

b.The ratio between the length of the car and the width of the car is not equal to 1.618.

c.The expectation of the ratio between the length of the car and the width of the car is equal to 1.618.

d.The expectation of the ratio between the length of the car and the width of the car is not equal to 1.618.

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It is assumed, when constructing the t-test, that the measurements are Normally distributed. In this exercise we examine the robustness of the test to divergence from the assumption. You are required to compute the significance level of a one-sided t-test of H₀: E(X) 5 versus H₁: E(X) < 5. Assume there are n = 35 observations and use a t-test with a nominal 5% significance level.

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Consider the test statistic T = [X-5]/[S/ n], whereXis the sample average, S²is the sample variance and n is the sample size. The structure of the rejection region of the test is:

Select one:

a.{|T| > c}

b.{T > c}

c.{T < c}

d.{T c}

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In order to obtain a significance level of 5% one should set the threshold c to be equal to: (Give an answer of the form x.xxx with 3 significance digits. Pay attention to the sign!)

Answer:

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Assume that the distribution of the observations is Binomial(20,0.25). Then the actual significance level of the test is (Use simulations to compute the significance level. Choose the closest answer.):

Select one:

a.0.01

b.0.03

c.0.05

d.0.07

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Assume that you are interested in testing H₀: E(X) 1 versus H₁: E(X) >1 using the t-test. Let the sample average, of a sample of size n = 220, be equal to x = 1.36 and the sample standard deviation be equal to s = 2.20.

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Thevalueof the test statistic is equal to: (Give an answer of the form x.xxx with 3 significance digits)

Answer:

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The p-value for testing thetwo-sidedhypothesis H₀: E(X) = 1 versus H₁: E(X) 1 using the t-test is equal to: (Give an answer of the form x.xxx with 3 significance digits)

Answer:

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Assume that you are interested in testing that the probability of an event is H₀: p 0.25 versus H₁: p > 0.25 using the test for proportions. Let the proportion of the event in a sample of size n = 2400 be equal to 0.275.

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The p-value for testing the given hypotheses is equal to: (Use the function "prop.test"witha continuity correction. You may use the argument "alternative="greater"". Give an answer of the form x.xxx with 3 significance digits)

Answer:

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In Question 11.1 of the Solved Exercises of Chapter 11 we introduced an experiment aim at assessing the effect of rumors and prior reputation of the instructor on the evaluation of the instructor by her students. The data produced by the experiment are stored in the file "teacher.csv". The file can be found on the internet atthis link. Use this data in order to answer the next 3 questions:

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We are interested in constructing a confidence interval for the ratio of the variance of the measurement in sub-population "a" divided by the variance of the measurement in sub-population "b". It is assumed in the construction that the measurements are Normally distributed. In the next 4 questions you are required to examine the robustness of confidence level to divergence from the assumption.

Assume there are n_a=29 observations in one group and n_b= 21 observations in the other group. Consider a confidence interval with a nominal 95% confidence level.

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If the distribution of the measurement in sub-population "a" is Uniform(0,4) and the distribution of the measurement in sub-population "b" is Uniform(0,5) then the actual value of the estimated parameter is:

Select one:

a.0

b.0.64

c.0.8

d.1

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Subpopulation "a" has a Uniform(0,4) distribution, and subpopulation "b" has a Uniform (0,5) distribution.

Use simulation, using sample sizes of 29 and 21 respectively,to find a sample of10^5 variances for

each of these subpopulations, and use these to find 10^5 values of

F = variance of "a" / variance of "b".

What proportion of these F-values falls in the middle .95 interval of the F distribution with the degrees offreedom appropriate for the sample sizes? Choose the closest answer.

Select one:

a.0.87

b.0.91

c.0.95

d.0.99

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If the distribution of the measurement in sub-population "a" is Poisson(4) and the distribution of the measurement in sub-population "b" is Poisson(6) then the actual value of the estimated parameter is:

Select one:

a.0

b.0.444

c.0.667

d.1

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The distribution of the measurement in sub-population "a" is Poisson(4) and the distribution of the measurement in sub-population "b"is Poisson(6). Using simulation, find 10^5 values of the variances and 10^5 values of F. What proportion of the F-values falls

in the middle 95% interval of the appropriate F distribution?Choose the closest answer.

Select one:

a.0.87

b.0.91

c.0.95

d.0.99

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The sample average in one sub-sample isx_a= 4.3 and the sample standard deviation is s_a= 5.4. The sample average in the second sub- sample isx_b= 2.1 and the sample standard deviation is s_b= 6.7. The size of the first sub-sample is n_a= 83 and the size of the second sub-sample is n_b= 92. We are interested in testing that the expectations in the associated sub-populations are equal to each other. The following 3 questions refer to the given information.

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Theabsolute valueof the test statistic is equal to: (Give an answer of the form x.xxx with 3 significance digits)

Answer:

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The p-value for testing the hypothesis is equal to: (Use the Normal approximation. Choose the closest answer.)

Select one:

a.0.15

b.0.04

c.0.015

d.0.004