2 (2) Consider the following univariate regression model:Y; = Bo+ B1D; + U;,where D; = 1 or 0. Then show that the least squares estimator of ß, takes the follow-ing form:E, Y;(1– D;)(1– D;) *i=1vi=1Zi=1

To derive the least squares estimator of in the given univariate regression model, we need to…

Answered: (2) Consider the following univariate…

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter3: Polynomial Functions

Section: Chapter Questions

Problem 18T

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Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?
bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.
The relationship between number of beers consumed (x) and blood alcohol content (y) was studied in 16 male college students by using least squares regression. The following regression equation was obtained from this study: y-hat = -0.0127 + 0.0180x. Wendall drinks 4 beers and has a blood alcohol level of .08. What is Wendall’s residual?
The accompanying data represent the weights of various domestic cars and their gas mileages in the city. The linear correlation coefficient between the weight of a car and its miles per gallon in the city is r= - 0.984. The least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable is y = - 0.0066x + 43.3954. Complete parts (a) and (b) below. Click the icon to view the data table. (a) What proportion of the variability in miles per gallon is explained by the relation between weight of the car and miles per gallon? Data Table The proportion of the variability in miles per gallon explained by the relation between weight of the car and miles per gallon is %. (Round to one decimal place as needed.) (b) Interpret the coefficient of determination. Full data set % of the variance in is by the linear model. Miles per Miles per Weight (pounds), x Weight (pounds), x Car Car (Round to one decimal place as needed.) Gallon, y Gallon, y…
Find the least-squares regression line treating square footage as the explanatory variable. y = (Round the slope to three decimal places as needed. Round the intercept to one decimal place as needed.)
The scatter plot below shows the average cost of a designer jacket in a sample of years between 2000 and 2015. The least squares regression line modeling this data is given by yˆ=−4815+3.765x. A scatterplot has a horizontal axis labeled Year from 2005 to 2015 in increments of 5 and a vertical axis labeled Price ($) from 2660 to 2780 in increments of 20. The following points are plotted: (2003, 2736); (2004, 2715); (2007, 2675); (2009, 2719); (2013, 270). All coordinates are approximate. Interpret the slope of the least squares regression line. Select the correct answer below: 1.The average cost of a designer jacket decreased by $3.765 each year between 2000 and 2015. 2.The average cost of a designer jacket increased by $3.765 each year between 2000 and 2015. 3.The average cost of a designer jacket decreased by $4815 each year between 2000 and 2015. 4. The average cost of a designer jacket increased by $4815 each year between 2000 and…
The relationship between a number of beers consumed (x) and blood alcohol content (y) was studied in 16 male college students by using least squares regression. The following regression equation was obtained from this study: ?̂ = -0.0127 + 0.0180x The above equation implies that: each beer consumed increases blood alcohol by 1.27% on average it takes 1.8 beers to increase blood alcohol content by 1% each beer consumed increases blood alcohol by an average of the amount of 1.8% each beer consumed increases blood alcohol by exactly 0.018
A study investigated how the content of vitamin A in carrots is affected by the time being cooked. In this example: X represents the amount of time, in minutes, that the carrot slices were cooked Y represents the content of vitamin A (in milligrams) in the carrot slices The least-squares regression equation for this relationship is: Y = 21.4 – 0.75X What is the slope of the regression line? Provide a numeric value as shown in the equation.
The regression equation of production (x) on capacity utilisation () of a certain firm is 3x – 2.76y – 102.30 = 0. The average capacity utilisation of the firm was 70% and the variance of capacity utilisation Is ) thof the variance of production. Find the average production and coefficient of correlation Between production and capacity utilisation.
Q5/ Use Linear Regression to fit the following data: X 1 2 4 5 6 Y 4 10 10 9 3
A recent study showed that the hours a person exercised in a week affected the individual'sresting heart rate. It was computed that r = -.68 and the least squares regression line was?̂ = 83-1.4x, where x is the hours exercised and y is the resting heart rate. d. What percentage of variability in resting heart rate can be explained by variability inhours exercised?
Suppose Wesley is a marine biologist who is interested in the relationship between the age and the size of male Dungeness crabs. Wesley collects data on 1,000 crabs and uses the data to develop the following least-squares regression line where X is the age of the crab in months and Y is the predicted value of Y, the size of the male crab in cm. Y = 8.2052 + 0.5693X What is the value of Ý when a male crab is 21.7865 months old? Provide your answer with precision to two decimal places. Interpret the value of Ý. The value of Ý is the probability that a crab will be 21.7865 months old. the predicted number of crabs out of the 1,000 crabs collected that will be 21.7865 months old. the predicted incremental increase in size for every increase in age by 21.7865 months. the predicted size of a crab when it is 21.7865 months old.

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(2) Consider the following univariate regression model: Y₁ = B₁ + B₁ D₁ + Uj, where D₁ = 1 or 0. Then show that the least squares estimator of ß₁ takes the follow- ing form: Σ²₁Y;D; n Σ₁₁ Di Σ²₁Y;(1 – D;) Σ(1-D;)