In a controlled experiment, the height Y in feet of a tree and the amount of water x in ounces received are measured. Find the least squares equation relating Y to x if we observe the following data for n=8 trees Σα; = 192, Σ, y = 258 Σ; 02; = 5280, Σ; €iyi = 6864 ŷ=1.29+1.29x ŷ=40.50+9.06x ŷ=8.40+1.3x y=8.25+x
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- A chemical company, seeking to study the effect of extraction time on the efficiency of an extraction operation, obtained the data from a table: Fit a straight line to the data given with the least squares method and use it to predict the extraction efficiency that would be expected when the extraction time is 35 minutes. x = Extraction time (minutes)y = Extraction efficiency (%)The following data is about the number of turns required to break a certain type of forged alloy bar, as well as the percentages of two alloying elements present in the metal. A) Fit a least squares regression plane and use its equation to estimate the number of turns required to break one of the bars, when x1 = 2.5 and x2 = 12 In the tabley = number of turnsx1 =% of element Ax2 =% of element BBased on the data presented in the table below, please calculate the values of b0 and b1 using OLS (ordinary least squares) and for the equation:qi = β0 + β1pi + ui. qi pi 1 2 2 3 3 5
- Suppose a least-squares regression line is given by y=4.302x−3.293. What is the mean value of the response variable if x=20? μy20=_______? (Round to one decimal place as needed.)A regression analysis between weight (y in pounds) and height (x in inches) resulted in following least squares line: y^= 120+5x. this implies that if the height is increased by 1 inch, the weight is expected ?Suppose 5 out of 25 data points in a weighted least-squares problem have a y-measurement that is less reliable than the others, and they are to be weighted half as much as the other 20 points. One method is to weight the 20 points by a factor of 1 and the other 5 by a factor of 1/2. A second method is to weight the 20 points by a factor of 2 and the other 5 by a factor of 1. Do the two methods produce different results? Explain.
- Ex[lain the Two Stage Least Squares Estimator?The y-interept bo of a least-squares regression line has a useful interpretation only if the x-values are either all positive or all negative. Determine if the statement is true or false. Why? If the statement is false, rewrite as a true statement.The following table shows the percent of water and the number of calories in various canned soups to which 100 g of water are added. (show all the necessary solution) Percent Water in Soups % Water Calories 83.3 28 92.3 26 91.9 39 89.4 57 89.5 57 90.5 36 91.9 32 91.7 32 a. Find the equation of the least squares line for the data. b. Use the equation in part a to find the expected number of calories in a soup that is 87% water. c. Determine the correlation coefficient of the data.
- The regional transit authority for a major metropolitan area wants to determine whetherthere is any relationship between the age of a bus and the annual maintenance cost. Asample of 10 buses resulted in the following data. Age of Bus (year) Maintenance Cost ($) 1 350 2 370 2 480 2 520 2 590 3 550 4 750 4 800 5 790 5 950 a. Develop the least squares estimated regression equation.b. Test to see whether the two variables are significantly related with α = .5.c. Did the least squares line provide a good fit to the observed data? Explain.d. Develop a 95% prediction interval for the maintenance cost for a specific bus thatis 4 years old.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.The data in the table represent the weights of various domestic cars and their miles per gallon in the city for the 2008 model year. For these data, the least-squares regression line is y=−0.006x+42.541. A twelfth car weighs 3,425pounds and gets 12 miles per gallon. (a) Compute the coefficient of determination of the expanded data set. What effect does the addition of the twelfth car to the data set have on R2? (b) Is the point corresponding to the twelfth car influential? Is it an outlier? Car Weight_(pounds),_x Miles_per_Gallon,_y1 3767 212 3989 193 3534 204 3175 225 2581 276 3736 207 2606…