Concept explainers
If there is at least one x value at which more than one observation has been made, there is a formal test procedure for testing
Versus
Ha: H0 is not true (the true regression function is not linear)
Suppose observations are made at x1, x2, …, xc. Let Y11, Y12, …,
The ni observations at xi contribute ni – 1 df to SSPE, so the number of degrees of freedom for SSPE is
The test statistic is F = MSLF/MSPE, and the corresponding P-value is the area under the
x | 110 | 110 | 110 | 230 | 230 | 230 | 360 |
y | 235 | 198 | 173 | 174 | 149 | 124 | 115 |
x | 360 | 360 | 360 | 505 | 505 | 505 | 505 |
y | 130 | 102 | 95 | 122 | 112 | 98 | 96 |
(So c = 4, n1 = n2 = 3, n3 = n4 = 4.)
a. Test H0 versus Ha at level .05 using the lack-of-fit test just described.
b. Does a scatterplot of the data suggest that the relationship between x and y is linear? How does this compare with the result of part (a)? (A nonlinear regression function was used in the article.)
Want to see the full answer?
Check out a sample textbook solutionChapter 13 Solutions
Probability and Statistics for Engineering and the Sciences
- The following fictitious table shows kryptonite price, in dollar per gram, t years after 2006. t= Years since 2006 0 1 2 3 4 5 6 7 8 9 10 K= Price 56 51 50 55 58 52 45 43 44 48 51 Make a quartic model of these data. Round the regression parameters to two decimal places.arrow_forwardA sample of the number of sales per hour at Store X and the number of sales at Store Y at the same times is as indicated below. Find the Y-value of the regression line where X = 5. X Y 0 1 0 2 1 2 3 4 4 6arrow_forwardGiven are five observations collected in a regression study on two variables. xi 2 6 9 13 20 yi 7 18 9 26 23 Compute b0 and b1 (to 1 decimal).b1 b0 Complete the estimated regression equation (to 1 decimal).^y = + x Use the estimated regression equation to predict the value of y when x = 6 (to 1 decimal).^y =arrow_forward
- For the following data: X Y 3 8 5 8 2 6 2 3 4 6 1 4 4 7 Find the regression equation for predicting Y from X. ∑X SSXX ∑Y ∑XY SP Ŷ = X Use the regression equation to find a predicted Y for each X. X Y Ŷ 3 8 5 8 2 6 2 3 4 6 1 4 4 7arrow_forwardIn a simple ridge regression model, if the (x, y) pairs (0, 1) and (1, 0) yield the same error factor (E or e) of value 2, which of the following may be (closest) used for the regularization parameter λ? a) λ = 0.0 b)λ = 0.5 c)λ = 1.0 d)λ = 1.5arrow_forwardFor 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)arrow_forward
- Consider the following population linear regression model of individual food expenditure: Y = 50 + 0.5X + u, where Y is weekly food expenditure in dollars, X is the individual’s age, and 50+0.5X is the population regression line. Suppose we generate artificial data for 3 individuals using this model. This artificial sample, which consists of 3 observations, is shown in the following table: Answer the following questions. Show your working. (a) What are the values of V1 and V4? (b) Suppose we know that in this artificial sample, the sample covariance between X and Y is 150, and the sample variance of X is 100. Compute the OLS regression line of the regression of Y on X. (Hint: Assume these summary statistics and the OLS regression line continue to hold in parts (c)-(e).) (c) What are the values of V5 and V7?arrow_forwardHello, please help me to solve the question below.Consider this regression model: Yt = β0 + β1 Ut + β2 Vt + β3 Wt + β4 Xt + εt ; where t= 1, ..., 75.We use OLS to estimate the parameters, producing the following model:Ŷt = 1.115 + 0.790 Ut − 0.327 Vt + 0.763 Wt + 0.456 Xt (0.405) (0.178) (0.088) (0.274) (0.017) Given that:R2 = 0.941; Durbin Watson stat DW = 1.907; RSS = 0.0757.(To answer the question, use the 5% level of significance, state clearly H0 and H1 that are tested, the test statistics that are used, and interpret the decisions.) (a) Describe the concepts of unbiasedness and efficiency. State the conditions required of regression (1) in order that the OLS estimators of the model parameters possess these properties. (b) Perform the following tests on the parameters of regression (1): (i) test whether the parameters β1, β2, β3 and β4 are individually statistically significant; (ii) test the overall significance of the regression model; (iii) test whether…arrow_forwardD2) Consider the following simple regression model y = β0 + β1x1 + u. Suppose Corr(x,u) > 0, Corr(z,x) > 0, and Corr(z,u) < 0. Then, the OLS estimator has a(n) _____. Select one: a. asymptotic bias b. upward bias c. downward bias d. substantial biaarrow_forward
- Consider the linear regression model y = β0 + β1x + u, where the regressor x may be correlated with the error term u. A poor instrumental variable for the regressor x would be one that a. predicts x. b. is not entirely related to x. c. predicts y after x is controlled for. d. None of the listed answers is correct. e. is uncorrelated with u.arrow_forwardSuppose that n = 50, i.i.d. observations for (Yi, Xi) yield the followingregression results: Ŷ= 49.2 + 73.9X, SER = 13.4, R2 = 0.78. (23.5) (16.4)Another researcher is interested in the same regression, but makes an errorwhen entering the data into a regression program: The researcher enterseach observation twice, ending up with 100 observations (with observation1 entered twice, observation 2 entered twice and so forth).a. Using these 100 observations, what results will be produced by theregression program? (Hint: Write the “incorrect” values of the samplemeans, variances, and covariances of Y and X as functions of the “correct”values. Use these to determine the regression statistics.) Ŷ = ____ + ____X, SER = ____, R2 = ____. (____) (____)b. Which (if any) of the internal validity conditions are violated?arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning