Problem 2. Mean Absolute Error - Adding a Data PointSuppose we have a dataset y₁ ≤ Y2 < ... < Yn and want to minimize absolute loss on it for the prediction h.As we've seen before, the corresponding empirical risk is mean absolute error,nRabs(h) =| Yi - hnSuppose that Rabs(a)=Rabs (α + 3)=a)b)M, where M is the minimum value of Rabs(h)and a is some constant. Suppose we add to the dataset a new data point yn+1 whose value is a + 1.For this new larger dataset: What is the minimum of Rabs(h)? At what value of h is this minimumachieved?Your should only involve the variables n, M, a, and constants.Let n be odd. Let ya and yɩ be two values in our dataset such that ya < yɩ and that the slopeof Rabs(h) is the same between hYa and h= y. Specifically, let d be the slope of Rabs (h) betweenYa and yb.=1=Suppose we introduce a new value q to our dataset such that q> y. In our new dataset of n + 1values, the slope of Rabs (h) is still the same between h== Ya and hYb, but it's no longer equal tod. What is the slope of Rabs (h) between h= Ya and h= y in our new dataset? Your answer shoulddepend on d, n, q, and/or one or more constants.

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Answered: Problem 2. Mean Absolute Error - Adding…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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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?
What does the y -intercept on the graph of a logistic equation correspond to for a population modeled by that equation?
Find the equation of the regression line for the following data set. x 1 2 3 y 0 3 4
10) The following results are from a regression where the dependent variable is COST OF COLLEGE and the independent variables are TYPE OF SCHOOL which is a dummy variable = 0 for public schools and = 1 for private schools, FIRST QUARTILE SAT which is the average score of students in the top quartile of SAT’s, THIRD QUARTILE SAT which is the average score of students in the 3rd quartile, and ROOM AND BOARD which is the cost of room and board at the school. The first set of results includes all the independent variables whereas the second set of results excludes the THIRD QUARTILE SAT variable. a) Based on these two sets of data, does there appear that multicollinearity is a problem (specifically, does it appear that THIRD QUARTILE SAT is highly collinear with the other independent variables? Explain. b) Calculate the VIF for THIRD QUARTILE SAT. c) Based on the VIF, do you think that multicollinearity is a problem? Explain.
2. Suppose two researchers independently collect samples of 100 house prices and their floor areas. They each run a regression of house price on floor area (including an intercept) and come up with two estimates of 3₁. Will those estimates be exactly equal? Explain. For now, assume homoskedasticity: Var (e|x) = 0².
Problem 2. Show that D₁ = (DFFITS;)² MSE (1) (p+1)MSE' where MSE() is the mean squared error after the i-th data point is omitted.
To estimate a population mean μ, the sample mean x¯ is often used as an estimator. However, a different estimator is called the sample midrange, given by the formula sample minimum + sample maximum/ 2. (a) The following table shows the values, in thousands of dollars, of 40 randomly selected houses in the city. (i) Calculate the sample midrange for the data. (ii) Explain why the sample midrange might be preferred to the sample mean as an estimator of the population mean. (b) To investigate the sampling distribution of the sample midrange, a simulation is performed in which 100 random samples of size n=40 were selected from the population of house values. For each sample, the sample midrange was calculated and recorded on the following dotplot. The mean of the distribution of sample midranges is $617,000 with standard deviation $136,000. Based on the results of the simulation, explain why the sample mean might be preferred to the sample midrange as an estimator of the population mean.
This table reports the regression coefficients when the returns of the size-institutionalownership portfolio (columns 1 and 2) returns are regressed on three variables: a constant(column 3), the stock market returns (column 4), and the change of the value weighted discountof the closed end fund industry (column 6). Columns 5 and 7 report the corresponding t-statistics of the coefficient estimates. Note that a t-statistic with an absolute value above 1.96means the coefficient estimate is significantly different from 0 at the 1% level. Column 8reports the R square of the regressions. Column 9 reports the mean institutional ownership ofeach portfolio. The last column reports the F-statistics for a multivariate test of the null hypothesis that the coefficient on ΔVWD in the Low (L) ownership portfolio is equal to theHigh (H) ownership portfolio. Two-tailed p-values are in parentheses. 1. What is the main finding of this Table? 2. What is the explanation for…
Dep. Variable: Model: Method: Date: Time: No. Observations: Df Residuals: Df Model: Covariance Type: Intercept educ IQ KWW exper black feduc meduc Least Squares Wed, 23 Nov 2022 08:57:40 coef -503.0058 40.5455 3.1654 7.2094 16.2338 -59.2987 10.8278 5.0471 std err nonrobust 138.947 8.660 1.199 wage OLS 2.120 3.740 52.143 5.353 6.106 722 714 7 R-squared: Adj. R-squared: Prob (F-statistic): F-statistic: Log-Likelihood: AIC: BIC: -3.620 4.682 2.640 3.400 4.340 -1.137 2.023 0.827 P> [t] 0.000 -775.800 0.000 0.008 0.001 0.000 0.256 0.043 0.409 0.189 0.181 23.83 23.543 0.812 3.047 8.891 -161.671 0.318 -6.941 3.310-29 -5288.5 (0.025 0.9751 1.0590+04 1.063e+04 -230.212 57.548 5.519 11.372 23.577 43.074 21.338 17.035 Above is a regression of men's monthly earnings in dollars as a function of their years of education, their IQ score in "points", their Knowledge of the World of Work (KWW) score in points (a measure of their knowledge of how occupations work), their years of work experience…
25. The following model was fitted to data on 32 insurance companies: 9 = 7.62 - 0.16xi + 1.23x2 R = 0.37 (0.008) (0.496) where 9 = price-earnings ratio XI = size of insurance company assets, in billions of dollars x2 = dummy variable taking the value I for regional companies and 0 for national companies The numbers in parentheses under the coefficients are the estimated coefficient standard errors. a. Interpret the estimated coefficient on the dummy variable. b. Test against a two-sided alternative. the null hypothesis that the true coefficient on the dummy variable is 0. c. Test, at the 5% level, the null hypothesis B, = B2 = 0, and interpret your result.
This dataset of size n - 51 is for the 50 states and the District of Columbia in the United States. The variables are - year 2002 birth rate per 1000 females 15 to 17 years old and z- poverty rate (the percent of the state's population living in households with in- comes below the federally defined poverty level) i- 13.1 - 22.3 R- 51, Su - E(-2) - 914.7 S, - E( -2)( - 6) - 1, 256.2 Su - E(m - M) - 3, 249 SSE - E(w - - 1, 509.6 1. Find A. A and r. 2. Write the fitted linear regression model for ý, in terms of z, A, and A- For A and , use the values you obtained in problem 1. 3. Interpret the slope of the least squares line based on the context. 4. Find SST, SE and SSR. 5. Find R* and . 6. Complete a chart for ANOVA df MS pvalue Source Regression Error Total 7. Complete a chart for Parameter Estimates. t-value pvalue Parameter Intercept Slope Estimate S.e. 8. What is the residual e, for the observation (.) - (20.1,31.5)? 9. What is the conclusion for the test Ha : -0 vs Ho : +0 with a -…
Please do 3c and 3d with handwritten working out

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