[SOLVED] stats500 -HOMEWORK  3

Description

Using the dataset gala we discussed in class. Consider a regression model with “Endemics” as the response and “Area”, “Elevation”, “Nearest”, “Scruz”, “Adjacent” as predictors.

1. • What would be the H₀ and H_A 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?
   • For the regression model above, find 99% confidence intervals for βElevation and βNearest,
   • For α = 0.05, conduct a test for H₀ β_Nearest = β_Scruz = 0. What would be the p-value for this test? Based on your analysis, do you feel any of these predictors have an effect on the response? Without drawing the 95% simultaneous confidence region for (β_Nearest,β_Scruz), 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 ***

—

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

• What would be the H₀ and H_A 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?
• 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 H_A: β_ratio 6= 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?