1. Suppose that ₁ and 2 are unbiased estimators of the parameter. If V (₁) = 10 and V(6₂) = 4. Which estimator is best and in what sense is it best? Calculate the relative efficiency of the two estimators.
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- 4. The following output from R presents the results from computing a linear model. In our data example we are interested to study the relationship between students' academic performance api00 with variable enroll which is the number of students in the school. Call: Im (formula = api00 enroll, data = d) Residuals: мin 10 Median 30 Маx -285.50 -112.55 -6.70 95.06 389.15 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 744.25141 15.93308 46.711 < 2e-16 *** enroll -0.19987 0.02985 -6.695 7.34e-11 *** --- Signif. codes: O ' *** ' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 135 on 398 degrees of freedom Multiple R-squared: 0.1012, Adjusted R-squared: F-statistic: 44.83 on 1 and 398 DF, p-value: 7.339e-11 0.098984. In multiple regression, why do we prefer our IVs to be uncorrelated with each other (i.e., we want rx, x, to be 0)?Q1 When regressing number of crimes committed by an individual against years of education you omit the relevant variable innate ability. Then, the estimated coefficient for education is going to be: A downwards biased B. upwards biased C. biased, but we cannot say in which direction D. not biased Q2 You estimate the equation: Y=6.2+0.41X. After finding heteroscedasticity, you run a second equation: Y/X=0.39+6.31/X. What is the effects of X on Y based on your preferred specification? A. A 1 unit change in X increases Y by 6.31 units. B. A 1 unit change in X increases Y by 0.39 units C. A 1 unit change in X increases Y by 6.31%. D. A 1 unit change in X increases Y by 39% Q 3 Which of the following are advantages of the use of panel data over pure cross- sectional or pure timeseries modelling? (i) The use of panel data can increase the number of degrees of freedom and therefore the power of tests (ii) The use of panel data allows the average value of the dependent…
- Heights (cm) and weights (kg) are measured for 100 randomly selected adult males, and range from heights of 135 to 188 cm and weights of 39 to 150 kg. Let the variable x be the height. The 100 paired measurements yield x=167.08 cm, y=81.32 kg, r=0.254, and y=−109+1.07x. Find the best predicted value of y (weight) given an adult male who is 163 cm tall. Use a 0.05 significance level. The best predicted value of y for an adult male who is 163 cm tall is kg. (Round to two decimal places as needed.)Consider the following data on the number of minutes (x) that 10 persons spent on social media during office hours and their productivity level (y): Xi 10 29 54 63 70 76 88 91 108 118 Yi 92 72 59 50 49 48 38 25 14 9 A linear model was fitted using the statistical software R, producing the following output: Coefficients: (Intercept) 98.81082 -0.75263 Estimate Std. Error t value Pr(>|t|) 29.8 1.74e-09 *** -17.6 1.1le-07 *** 3.31574 0.04277 Signif. codes: 0 ***' 0.001 **** 0.01 0.05 .' 0.1 1 Residual standard error: 4.304 on 8 degrees of freedom Multiple R-squared: 0.9748, F-statistic: 309.7 on 1 and 8 DF, Adjusted R-squared: p-value: 1.11e-07 0.9717 (a) Obtain the equation of the estimated regression line. (b) Interpret the slope coefficient. (c) Discuss whether the simple linear regression model obtained does a good job of explaining observed variation in productivity level. (d) Perform a model utility test using a = 0.01. Use an appropriate P-value from the output given. (Note: R uses…(b) For a given set of trivariate data, the following results were obtained: X = 53, Y = 28, 6 (Y on X) =- 1.5, d (X on Y)=- 0.2. Find : (i) The two regression equations. (ii) the coefficient of correlation. (iii) The most probable value of Y when X=60.
- The research hypothesis posits that the treatment association between two variables will be greater than zero (H1: rxy > 0). What would be concluded for r(29) = .267, p > .05?Q3. A certain stimulus is to be tested for its effect on blood pressure. Twelve men have their blood pres- sure measured before and after the stimulus. The results are Man 1 3 4 6 7 8. 9. 10 11 12 Before (X) 120 124 130 118 140 128 140 135 126 130 126 127 After (Y) 128 131 131 127 132 125 141 137 118 132 129 135 In this particular scenario, we define W = Y – X as the improvement or increase in blood pres- sure due to the stimulus and assume that W;'s (i = 1, 2, ..., 12) are independent and distributed accordingi to a n(µw,ow) distribution. Let E(X)= µx and E(Y) = µy. (a) Construct a 95% confidence interval for x – µy, the average effect of the stimulus on blood pressure. Do you think the stimulus does have an effect on the blood pressure based on this 95% confidence interval? Briefly explain your answer. (b) Construct a 95% confidence interval for ow.8. In an investigation to empirically determine the value of 1, a student measures the circumference and diameter of several circles of varying size and uses Excel to make a linear plot of circumference versus diameter (both in units of meters). A linear regression fit yields the result of: y = 3.1527x - 0.0502, with R2= 0.9967 for the 5 data points plotted. How should this student report the final result? Does the empirical ratio of C/D agree with the accepted value of n?
