STATS500 Homework 3 Solved

1. Using the dataset gala we discussed in class. Consider a regres- sion model with “Endemics” as the response and “Area”, “Elevation”, “Nearest”, “Scruz”, “Adjacent” as predictors.
   1. (a)  What would be the H0 and HA if you wish to claim that an island with a large highest elevation level tends to have more endemic species. What would be the test statistic, p-value and conclusion for your test if α-level is 0.01?
   2. (b)  For the regression model above, find 99% confidence intervals for βElevation and βNearest, respectively.
   3. (c)  For α = 0.05, conduct a test for H0 βNearest = βScruz = 0. What would be the p-value for this test? Based on your anal- ysis, do you feel any of these predictors have an effect on the response? Without drawing the 95% simultaneous confidence region for (βNearest,βScruz), please make a guess whether (0,0) would be inside this confidence region or not. Briefly explain your answer.
2. Use the sat data (see help(sat) for the description of variables). Fit a model with total sat score as the response and takers, ratio and salary as predictors. Answer the following question using the output provided here:

       > var(sat$total)
       [1] 5598.116
       > tmp=lm(total~takers+ratio+salary, sat)
       > summary(tmp)
       Call:
       lm(formula = total ~ takers + ratio + salary, data = sat)
       Residuals:
   

        Min      1Q  Median      3Q     Max
    -89.244 -21.485  -0.798  17.685  68.262
    Coefficients:

                 Estimate Std. Error t value Pr(>|t|)
    (Intercept) 1057.8982    44.3287  23.865   <2e-16 ***
    takers        -2.9134     0.2282 -12.764   <2e-16 ***

ratio         -4.6394     2.1215  -2.187   0.0339 *
  salary         2.5525     1.0045   2.541   0.0145 *
  ---
  Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
  Residual standard error: 32.41 on 46 degrees of freedom
  Multiple R-squared:  XX

1. (a)  What would be the H0 and HA if you wish to claim that a higher value of average pupil/teacher ratio (ratio) tends to lead to a lower sat score. What would be the numerical value of the test statistic, p-value and conclusion for your test if α-level is 0.01?
2. (b)  Let σ2 denote the variance of random errors in the regression model (model tmp in R) , based on the output, what should be the estimates of σ2 and R2 (XX value in Multiple R-squared) – you do not need to carry out the calculation but make sure that I can get the correct numbers using your answers and a plain calculator.

3. Use the sat data and fit a model with total sat score as the response and takers, ratio and salary as predictors. Let α = .05. Conduct a test with HA: βratio ̸= 0 by using a permutation test and report the testing result. Using the same permutation outcomes, what would be the p-value for the test you consider in Problem 2(a) above?