3470-Practice_Questions_Final

STAT 3470-AU 23 Practice Questions - Final November 29, 2023 Nasser Sadeghkhani Q.1 Regression methods were used to analyze the data from a study investigating the relationship between roadway surface temperature in F ( x ) and pavement defection ( y ). Summary quantities were n = 20, X y i = 12 . 75 , X y 2 i = 8 . 86 , X x i = 1478 , X x 2 i = 143 , 215 . 8 X x i y i = 1083 . 67 . (a) Calculate the least squares estimates of the slope and intercept. Estimate σ 2 . (b) Use the equation of the fitted line to predict what pavement deflection would be observed when the surface temperature is 90F. (c) Give a point estimate of the mean pavement deflection when the surface is 85F. (d) What change in mean pavement deflection would be expected for a 1F change in surface temperature? (e) Test for significance of regression using α = 0 . 05. What conclusion can you draw? (f) Estimate the standard errors of the slope and intercept. (g) Find a 95% CI for β 0 , and β 1 . (h) Find a 95% CI for expected value of pavement deflection (or true value of Y ) when the surface tem- perature is 85F. (i) Find 95% PI when the for the future pavement deflection when the surface temperature is 85F. (j) Complete the ANOVA table. What are your (null and alternative) hypotheses here. (k) What is your conclusion in (j). 1

Q.2 The number of emails arriving at a server per minute is claimed to follow a Poisson distribution . To test this claim, the number of emails arriving in 70 randomly chosen 1-minute intervals is recorded. The table below summarises the results. Test the hypothesis that the number of emails per minute follows a Poisson distribution? Use α = 0 . 05. # emails freq. 0 13 1 22 2 23 3 12 ≥ 4 0 2

Q.4 A manager in a trucking company wants to predict the total daily travel time for the drivers. He believes that total travel time (hours) depends of number of miles traveled in making deliveries x 1 , and number of deliveries x 2 . A random sample of 10 driving assignment are taken. Below is the part of Minitab output (missing values are denoted by ? ) The regression equation is Time = - 0.869 + 0.0611 Miles + 0.923 Deliveries Predictor Coef SE Coef T P Constant -0.8687 0.9515 -0.91 0.392 Miles 0.061135 0.009888 6.18 0.000 Deliveries 0.9234 0.2211 4.18 0.004 R-Sq = 90.4% R-Sq(adj) = 87.6% Analysis of Variance SOURCE DF SS MS F p Reg. ? 21.601 ? ? 0.000 Error. ? ? ? Total. ? ? (a) Find the estimated regression equation? (b) Predict the travel time when miles traveled x 1 and number of deliveries x 2 are 80, and 4 respectively. (c) interpret the estimated coefficient for Miles , i.e. 0 . 061135. (d) Complete the ANOVA table by finding missing values are denoted by ? 4

Q.5 Below is a survey of average used vehicle prices in American in 1957. X = Vehicle age (years) 1 2 3 4 5 6 7 8 9 10 Y = Average price ( $ ) 2651 1943 1494 1087 765 538 484 290 226 204 The scatter plot of the raw data Y and X (left), and a log transformation on Y (log( Y )) and X . (right), are depicted below. Below is the R summary of the regression of Y on X : Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2371.47 210.25 11.279 3.43e-06 *** age -255.14 33.89 -7.529 6.74e-05 *** --- Residual standard error: 307.8 on 8 degrees of freedom Multiple R-squared: 0.8763, Adjusted R-squared: 0.8609 Below is the R summary of the regression of log( Y ) on X : Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 8.164585 0.057051 143.11 6.36e-15 *** age -0.297680 0.009195 -32.38 9.03e-10 *** --- Residual standard error: 0.08351 on 8 degrees of freedom Multiple R-squared: 0.9924, Adjusted R-squared: 0.9915 5

Q.6 Use the ANOVA procedure to test if the linear regression if significant for the following table. compare test statistic with critical value. Set α = 0 . 05. 7