(X1, X2,..., Xn) are independent and identically distributed random variables with density function f ((x|0) = ÷e¯. Find the MLE of 0. %3D 20 7.
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- In a regression based on 30 annual observations, U.S. farm income was related to four independent variables—grain exports, federal government subsidies, population, and a dummy variable for bad weather years. The model was fitted by least squares, resulting in a Durbin-Watson statistic of 1.29. The regression of e2i on ŷi yielded a coefficient of determination of 0.043.a. Test for heteroscedasticity.b. Test for autocorrelated errors.A financial analyst is examining the relationship between stock prices and earnings per share. She chooses sixteen publicly traded companies at random and records for each the company's current stock price and the company's earnings per share reported for the past 12 months. Her data are given below, with x denoting the earnings per share from the previous year, and y denoting the current stock price (both in dollars). Based on these data, she computes the least-squares regression line to be ŷ =−0.2390+0.044x. This line, along with a scatter plot of her data, is shown below.An owner of a home in the Midwest installed solar panels to reduce heating costs. After installing the solar panels, he measured the amount of natural gas used y (in cubic feet) to heat the home and outside temperature x (in degree-days, where a day's degree-days are the number of degrees its average temperature falls below 65° F) over a 23-month period. He then computed the least-squares regression line for predicting y from x and found it to be ŷ = 85 + 16x. The software used to compute the least-squares regression line for the equation above says that r2 = 0.98. This suggests which of the following? 1. Gas used increases by square root of 0.98 = 0.99 cubic feet for each additional degree-day? 2. Although degree-days and gas used are correlated, degree-days do not predict gas used very accurately. 3. Prediction of gas used from degree-days will be quite accurate.
- A random sample of size n = 4 taken from a normal population with σ2 = 9 is used to test H0 : µ = µ0against H1 : µ = µ1 where µ1 > µ0. e null hypothesis will be rejected if X > µ ¯0 + 2.5. Find the levelof significance of the critical regionSuppose that the sales of a company (Y) is regressed on advertising expenditure (x) and labor cost (z), and the estimated regression equation is Y = 5 + 0.5x + 0.7z + u (where u is the error term). Here, sales, advertising expenditure and labor cost are measured in million Tk. Standard error for the coefficient of x is 0.04, standard error for the coefficient of z is 0.01, and the sample size is 20. Can we conclude that advertising expenditure is a statistically significant variable?A scientific foundation wanted to evaluate the relation between y= salary of researcher (in thousands of dollars), x1= number of years of experience, x2= an index of publication quality, x3=sex (M=1, F=0) and x4= an index of success in obtaining grant support. A sample of 35 randomly selected researchers was used to fit the multiple regression model. Parts of the computer output appear below. The least squares line fitted to the data is: salary = 2.001 + 0.33 x1 + 0.04 x2 + 0.69 x3 + 0.30 x4 + ε salary = 17.85 + 1.10 x1 + 0.32 x2 + 1.59 x3 + 1.29 x4 + ε salary = 2.001 + 0.33 x1 + 0.04 x2 + 0.69 x3 + 0.30 x4 salary = 17.85 + 1.10 x1 + 0.32 x2 + 1.59 x3 + 1.29 x4
- A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.23. A dietician is asked to test this claim and finds that a random sample of 24 servings has a variance of 1.27. At α=0.10, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed. (b) Find the critical value(s). I do not know how to calculate the critical value with my calculator. I have a TI83+.Are the following statements true or false? Explain your answer.a. “An ordinary least squares regression of Y onto X will not be internallyvalid if Y is correlated with the error term.”b. “If the error term exhibits heteroskedasticity, then the estimates of Xwill always be biased.”A chemical engineer discovered that by adding different amounts of an additive to gasoline, he could reduce the amount of nitrous oxides (NOx) from a car engine. A specified amount was added to one gallon of gasoline and the total amount of NOx in the exhaust was collected. Assume, in suitable units, that the data is: amount of additive 1 2 3 4 5 Nox 19 17 14 13 12 a) Obtain the least squares fit, of a straight line, for the amount of NOx b) Test whether or not the slope is β = 0. Consider a significance level of 0.01. c) Provide a 99% confidence interval for the mean value of NOx when the amount of additive is 9.
- (a) Find the equation of the least-squares line for the data. (Round all numerical values to two decimal places.) y= ?Prove the result that the R^2 associated with a restricted least squares estimator is never larger than that associated with the unrestricted least squares estimator. Conclude that imposing restrictions never improves the fit of the regression.A study was conducted to assess the relationship between students’s score in final exam (y) and number of hours spent for exam (x) in each day. Data on a random sample 20 students were obtained and a regression model was estimated; and the least squares estimates obtained are: intercept a=28.5 and slope b=4.3 with SE(b)=Sb=0.017. The SS are: TSS=2540 and ESS=850. ****** QA) What is the difference between exam score obtained by two students one who studied 5 hours and the other who studied 9 hours per day. QB) In the above Question 1, find 95% CI for the slope and interpret it. In the above Question 1, find and interpret the coefficient of determination (r-square value).