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STAT 501 Mid-Term Exam 2 Spring 2015

A mid-term exam assessing statistical analysis and hypothesis testing.

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    STAT 501 – Mid - Term Exam 2 – Spring 2015 – Due April 12 Instructions : Use Word to type your answers within this document. Then, submit your answers in the appropriate dropbox in ANGEL by the due date and within 3 hours of downloading the exam . The point distribution is located next to each question. 1. ( 4 x2 = 8 points) State which of the following statements is TRUE and which is FALSE. For the statements that are false , explain why they are false . a. R emov ing an outlier in a regression analysis will result in narrower confidence interval s . b. In a simple linear regression (SLR) model, if a log transformation is performed on X to remedy some non - linearity, the mean value of Y is bound to change. c. In model selection, t he high e st adjusted R 2 - value and the smallest S - value criteria always yield the same "best" models. d. Regression models with different r esponses, but the same predictor X matrix, will have the same leverage values. 2. (3+3+4+4+3+3 = 20 points) Open the “ Salary Data . ” The data set consis t s of current salaries (Salary in thousands of dollars ) fo r 63 individuals with information about their years of work experience ( YrsExp ) and highest degree attained (Degree) . Your goal is to fit a regression model to express the dependence of Y ( Salary ) on X ( YrsExp ) and Degree . a. Clearly d efine a set of indicator variables that could be used in a regression model to represent the qualitative variable Degree . [Hint: T hink carefully about the number of indicator variables needed given the number of levels of Degree and use “Bachelor” as the reference level. ] b. Write a population multiple linear regression equation for predictin g the current salary in terms of YrsExp and Degree . Since education leve l could impact th e dependence of Y on X, t he model should contain an interaction ef fect between YrsExp and Degree , together with their main effects. [Hint: Yo ur equation should include Y, X, the indicator variables you defined in part (a) , interaction terms, and population regression coefficients ( β ’s). ] c. Conduct a hypothesis t est for whether the average annual salary increase per year of experience differs by level of education (i.e., test if the slopes for two or more Degree categories differ ) . Write out the null and alternative hypotheses, the test statistic, the p - value, and the conclusion. [Minitab v17: Select S alary as the Response, YrsExp as the Continuous predictor , Degree as the categorical predictor, click “Model,” select both YrsExp and Degree together in the Predictors box and click the Add button next to “Interactions through order 2 . ” Minitab v16: Create interaction terms using Calc > Calculator before fitting the regression model. ] d. Write a new population regression equation based on your conclusion to part (c). Fit this model and conduct two separate hypothesis t est s for whether the mean salary for a fixed number of years’ experience differs by education level . For each test, write out the null and alternative hypotheses, the test statistic, the p - value, and the conclusion.

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    e. Based on your conclusion to part (d) , write th r e e fitted sample regression equation s that can be used to predict the current salary for each education level . [Hint: Y our equations should include number values, not β ’s. ] f. Based on one of the equation s from part (e), p redict the current salary of a PhD degree holder with 10 year s of work experience . [Hint: A point estimate is sufficient so there is no need for an interval .] 3. (4x2 = 8 points) Consider the following four graphs where the vertical axis represents Y and the horizontal axis represents X. Choose the most appropriate plot for each of the following models (where D 1 and D 2 represent a set of indicator variables) : a. 𝐸 ( 𝑌 ) = 𝛽 0 + 𝛽 1 𝑋 b. 𝐸 ( 𝑌 ) = 𝛽 0 + 𝛽 1 𝑋 + 𝛽 4 𝐷 1 𝑋 + 𝛽 5 𝐷 2 𝑋 c. 𝐸 ( 𝑌 ) = 𝛽 0 + 𝛽 1 𝑋 + 𝛽 2 𝐷 1 + 𝛽 3 𝐷 2 d. 𝐸 ( 𝑌 ) = 𝛽 0 + 𝛽 1 𝑋 + 𝛽 2 𝐷 1 + 𝛽 3 𝐷 2 + 𝛽 4 𝐷 1 𝑋

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    4. ( 5+2+5+3+3 = 18 points) The file “ Savings Data ” contains savings of 33 individuals along with their age . It is apparent that Y = S avings (in $) has a positive association with X = A ge (in years) . An appropriate regression model relating S avings to A ge could be useful for predicting savings based on age. The most straightforward approach would be to fit a simple linear regression (SLR) model for Y vs X, provided that the LINE assumptions are satisfied. [Consult “Worked Examples Using Minitab” in the Online Notes for help with any Minitab procedures.] a. Fit an SLR model for Y vs X and p erform a residual plot analysis to determine if the LINE assumptions are satisfied. Include a numerical test when checking for normality (u se t he Ryan Joiner test in Minitab). Discuss your findings and include any relevant graphs. b. Based on your conclusion in part ( a ) , determine if any transformations are suggested for X and/or Y. [Hint: You should find that both X and Y need to be transformed.] c. Fit an SLR model for the transformed variable ( s ) and comment on this model’s validity with supporting statements, numerical tests and/or plots . d. Use Minitab to compute a 95% confidence interval for the mean amount of savings (in $) expected for 40 year - olds based on the fitted model in part ( c ) . [Hint: R emember to take into account the transformations to X and Y.] e. Use Minitab to compute a 95% prediction interval for the amount of savings (in $) predicted for a randomly selected 40 year - old based on the fitted model in part ( c ) . [Hint: R emember to take into account the transformations to X and Y.] 5. (2+1+3+2 = 8 points) The following Minitab output resulted from a multiple linear regression model fit to response variable, Y , and predictor terms, X 1 , X 2 , and X 1 X 2 : Coefficients Term Coef SE Coef T - Value P - Value Constant 4.49 1.89 2.37 0.022 X1 0.759 0.374 2.03 0.048 X2 0.965 0.426 2.26 0.028 X1*X2 0.1742 0.0821 2.12 0.039 a. Conduct a hypothesis test for whether the interaction term, X 1 X 2 , can be dropped from the model . Write out the population model, null and alternative hypotheses, the test statistic, the p - value, and the conclusion. In this case the population model is, 𝑌 = 𝛽 0 + 𝛽 1 ∗ 𝑋 1 + 𝛽 2 ∗ 𝑋 2 + 𝛽 3 ∗ 𝑋 1 𝑋 2 + 𝑒 Here we want to test whether the interaction term is significant or not thus the null and alternative hypotheses are, 𝐻 0 : 𝛽 3 = 0 𝑎𝑛𝑑 𝐻 𝑎 : 𝛽 3 ≠ 0 From the above output we can see that the test statistic for this above hypothesis test is 2.12 with the corresponding p - value 0.039. Now assuming

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