Consider the simple linear regression model where Y1, . . . , Yn are independent random variables with Yi ∼ N(µi , σ 2 ), µi = β0 + β1xi , for i = 1, . . . , n. Given that we observe y1, . . . , yn, find the MLEs for β0, β1 and σ 2 .
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Consider the simple linear regression model where Y1, . . . , Yn are independent random variables with Yi ∼ N(µi , σ 2 ), µi = β0 + β1xi , for i = 1, . . . , n. Given that we observe y1, . . . , yn, find the MLEs for β0, β1 and σ 2 .
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- The following table provides values of the function f(x,y). However, because of potential; errors in measurement, the functional values may be slightly inaccurately. Using the statistical package included with a graphical calculator or spreadsheet and critical thinking skills, find the function f(x,y)=a+bx+cy that best estimate the table where a, b and c are integers. Hint: Do a linear regression on each column with the value of y fixed and then use these four regression equations to determine the coefficient c. x y 0 1 2 3 0 4.02 7.04 9.98 13.00 1 6.01 9.06 11.98 14.96 2 7.99 10.95 14.02 17.09 3 9.99 13.01 16.01 19.02In some situations where the regression model is useful, it is known that the mean of Y when X = 0 is equal to 0, that is, Yi = βxi +ǫi where ǫi for i = 1, 2, . . . , n are independent and N (0, σ2 ). Suppose a sample of size n = 2 consists of the observations (x1, y1) = (1, 1) and (x2, y2) = (2, 1). Calculate the estimates of β and σ 2 from part (1) for these data. Please step by step answer.Consider a simple linear regression model, Y₁ = Bo + B₁X₁ + ₁ for i=1,2,...,n with the usual random error term (&) assumptions. Suppose that hypotheses Ho: P₁ = 0 against H₁:₁ <0 were tested and Ho: B₁ = 0 was rejected. Assume ₁e₁² = 0. ei i) Briefly explain what the conclusion of the test means for the values of Y given the values of X. -1 ii) Draw an example of a scatter plot with the fitted regression line for the problem mentioned above. iii) Calculate and interpret the correlation coefficient for the problem at hand.
- Find the simple regression line y=α+βx for the pairs of points belonging to the independent and dependent variables (xi,yi) , respectively. Also, interpret the result by calculating the Pearson correlation coefficient.The number of murders and robberies per 100,000 population for a random selection of states are shown. Find the equation of the regression line y'=ax+b, and predict the number of robberies when x=4.5 murders murders,x: 2.4,2.7,5.6,2.6,2.1,3.3,6.6,5.7 robberies,y : 25.3,14.3,151.6,91.1,80,49,17.3,45.8(Yi, X1i, X2i) satisfy the assumptions. You are interested in β1, the causal effect of X1 on Y . Suppose that X1 and X2 are uncorrelated. You estimate β1 by regressing Y onto X1 (so that X2 is not included in the regression). Does this estimator suffer from omitted variable bias? Explain.
- Consider a set of data (x₁, y₁), i = 1, 2, ..., n and the simple linear regression model given by: Yi Bo + B₁x₁ + εi with E() = 0, Var(ɛ;) = o² and all &;'s are independent and normally distributed. Suppose further that a random sample of size 10 is selected and the fitted regression equation is obtained: with SB₁ b) = 0.071 and SSxx = 1568.625. a) If we were to draw a scatter plot for the data, what kind of relationship would be expected? Justify your answer. Construct analysis of variance (ANOVA) and use it to test at 5% level whether or not there is a significant relationship between x and y. Verify the results in b) using a 95% confidence interval. c) ŷi = 2.106 -0.933x₁Consider the model Ci= B0+B1 Yi+ ui. Suppose you run this regression using OLS and get the following results: b0=-3.13437; SE(b0)=0.959254; b1=1.46693; SE(b1)=0.0697828; R-squared=0.130357; and SER=8.769363. Note that b0 and b1 the OLS estimate of b0 and b1, respectively. The total number of observations is 2950. The number of degrees of freedom for this regression is A. 2950 OB. 2948 OC. 2952 OD. 2A statistics student is asked to estimate Y = Bo + B1X + E. She calculates the following values: Ex = 280, Σ(x₁ - x)² = 350, Σy, = 600, Σ(y, − y) = 1000 Σ(x,x)(y₁ - y) = -630, n = 20 Which of the following is the sample regression equation? OY--55.2-1.8X + e OY 55.2 +1.8X + e OY-55.2+1.8X + e OY 55.2 1.8X + e
- Find the simple regression line y=α+βx for the pairs of points belonging to the independent and dependent variables (xi,yi) , respectively. Also, interpret the result by calculating the Pearson correlation coefficient. Please dont use excel.Suppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 12 members of the population, where for each member, the value of Y as well as the values of the three explanatory unknowns were observed. The data is entered into a computer using linear regression software and the output summary tells us that R-square is 0.85, the linear model coefficient of the first explanatory unknown is 7 with standard error estimate 2.5, the coefficient for the second explanatory unknown is 11 with standard error 2, and the coefficient for the third explanatory unknown is 15 with standard error 4. The regression intercept is reported as 28. The sum of squares in regression (SSR) is reported as 85000 and the sum of squared errors (SSE) is 15000. From this information, what is SSE/SST? (a) .2 (b) .13 (c) NONE OF THE OTHERS (d) .15 (e) .25For x={1 2 3 4 5} and y={2 1 4 3 6} use normal equation (c =(ATA)-1ATy) to find with: a-) linear regression coefficients, b-) the linear regression equation, c-) residel sum of squares(RSS)