Question

"By Hand" Problems: For hypothesis tests, you may use R to find the p-value. For confidence intervals, you may use R to find the multiplier 1. (Continuation of HW 7, Problem 3) Suppose it is of interest to examine the relationship between the size of cruise chips and their passenger capacity. A data from 158 cruise ships was collected based on the following two variables: . Size of the ship (Tonnage) the gross tonnage in 1000s of tons Maximum passenger capacity (Passengers) maximum number of passengers in 100s of people A regression analysis in R was performed with the partial results given below along with a scatterplot of the data. > cruise.mod 1m (Passengers Tonnage) summary (cruise.mod) Coefficients: Estimate Std. Error t value Pr>ltI) (Intercept) 0.946302 0.546733 1.731 0.0855 Tonnage 0.245650 0.006803 36.109 <2e-16 Signif. codes: 0.001 0.010.05 0.1 1 Residual standard error 3.173 on 156 degrees of freedom Multiple R-squared: 0.831 Adjusted R-squared: 0.8925 F-statistic: 1304 on 1 and 156 DF, p-value: < 2.2e-16 Use the partial R output above to answer the following questions (a) Obtain a point estimate of ơ2 from the output (b) Compute and interpret a 95% confidence interval for the slope. (c) Is there evidence that the size of the ship is linearly related to the number of passengers? Use the R output to conduct the appropriate hypothesis test. Show all five steps as usual

Answer 1

$(a)~\hat{\sigma}^2=3.173^2=10.06793 \\ (b)~95\%~C.I.~for~slope,~\beta:\\ Lower~limit=\hat{\beta}-t_{0.025,156}se(\hat{\beta})=0.245650-1.9753*0.006803=0.232212 \\ \hat{\beta}=0.245650,~t_{0.025,156}=1.9753,~se(\hat{\beta})=estimate~of~standard~error~of~\hat{\beta}=0.006803\\ Upper~limit=\hat{\beta}+t_{0.025,156}se(\hat{\beta})=0.245650+1.9753*0.006803=0.259088\\ We~are~95\%~confident~that~the~true~value~of~slope~lies~between~\\ 0.006803~and~0.259088.\\ (c)~Step~1:\\ Null~hypothesis,~H_0:~size~of~the~ship~is~not~linearly~related~to~the~number~of\\ passengers~i.e.~\beta=0\\ Alternative~hypothesis,~H_1:~size~of~the~ship~is~linearly~related~to~the~number~of\\ passengers~i.e.~\beta\neq 0\\ Value~of~F=1304~(or~equivalent~test~statistic,~value~of~t=36.109)\\ Critical~value=F_{0.05,1,156}=3.9018\\ Since~p-value~corresponding~F-statistic=2.2\times 10^{-16}<0.05~(or,~value~of~F>3.9018)~so\\ we~reject~H_0\\ We~conclude~that~the~size~of~the~ship~is~linearly~related~to~the~number~of\\ passengers.$