Problem 13
Test the correlation, as indicated. Show all details of the test. Test for evidence of a linear association; \(r=0.28 ; n=100\).
Problem 14
Test the correlation, as indicated. Show all details of the test. Test for a negative correlation; \(r=-0.41\); \(n=18\).
Problem 23
Hantavirus is carried by wild rodents and causes severe lung disease in humans. A study \(^{5}\) on the California Channel Islands found that increased prevalence of the virus was linked with greater precipitation. The study adds "Precipitation accounted for \(79 \%\) of the variation in prevalence." (a) What notation or terminology do we use for the value \(79 \%\) in this context? (b) What is the response variable? What is the explanatory variable? (c) What is the correlation between the two variables?
Problem 24
Teams in the National Football League (NFL) in the US play four pre-season games each year before the regular season starts. Do teams that do well in the pre-season tend to also do well in the regular season? We are interested in whether there is a positive linear association between the number of wins in the pre-season and the number of wins in the regular season for teams in the NFL. (a) What are the null and alternative hypotheses for this test? (b) The correlation between these two variables for the 32 NFL teams over the 10 year period from 2005 to 2014 is 0.067 . Use this sample (with \(n=320\) ) to calculate the appropriate test statistic and determine the p-value for the test. (c) State the conclusion in context, using a \(5 \%\) significance level. (d) When an NFL team goes undefeated in the pre-season, should the fans expect lots of wins in the regular season?
Problem 27
A random sample of 50 countries is stored in the dataset SampCountries. Two variables in the dataset are life expectancy (LifeExpectancy) and percentage of government expenditure spent on health care (Health) for each country. We are interested in whether or not the percent spent on health care can be used to effectively predict life expectancy. (a) What are the cases in this model? (b) Create a scatterplot with regression line and use it to determine whether we should have any serious concerns about the conditions being met for using a linear model with these data. (c) Run the simple linear regression, and report and interpret the slope. (d) Find and interpret a \(95 \%\) confidence interval for the slope. (e) Is the percentage of government expenditure on health care a significant predictor of life expectancy? (f) The population slope (for all countries) is 0.467 . Is this captured in your \(95 \%\) CI from part (d)? (g) Find and interpret \(R^{2}\) for this linear model.
Problem 28
A common (and hotly debated) saying among sports fans is "Defense wins championships." Is offensive scoring ability or defensive stinginess a better indicator of a team's success? To investigate this question we'll use data from the \(2015-2016\) National Basketball Association (NBA) regular season. The data \(^{6}\) stored in NBAStandings2016 include each team's record (wins, losses, and winning percentage) along with the average number of points the team scored per game (PtsFor) and average number of points scored against them ( PtsAgainst). (a) Examine scatterplots for predicting \(\operatorname{WinPct}\) using PtsFor and predicting WinPct using PtsAgainst. In each case, discuss whether conditions for fitting a linear model appear to be met. (b) Fit a model to predict winning percentage (WinPct) using offensive ability (PtsFor). Write down the prediction equation and comment on whether PtsFor is an effective predictor. (c) Repeat the process of part (b) using PtsAgainst as the predictor. (d) Compare and interpret \(R^{2}\) for both models. (e) The Golden State Warriors set an NBA record by winning 73 games in the regular season and only losing 9 (WinPct \(=0.890\) ). They scored an average of 114.9 points per game while giving up an average of 104.1 points against. Find the predicted winning percentage for the Warriors using each of the models in (b) and (c). (f) Overall, does one of the predictors, PtsFor or PtsAgainst, appear to be more effective at explaining winning percentages for NBA teams? Give some justification for your answer.
Problem 30
We show an ANOVA table for regression. State the hypotheses of the test, give the F-statistic and the p-value, and state the conclusion of the test. $$ \begin{array}{l} \text { Analysis of Variance } \\ \begin{array}{lrrrr} \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Regression } & 1 & 303.7 & 303.7 & 1.75 & 0.187 \\ \text { Residual Error } & 174 & 30146.8 & 173.3 & & \\ \text { Total } & 175 & 30450.5 & & & \end{array} \end{array} $$
Problem 31
We show an ANOVA table for regression. State the hypotheses of the test, give the F-statistic and the p-value, and state the conclusion of the test. $$ \begin{array}{lrrrr} \text { Analysis of Variance } & & & & \\ \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Regression } & 1 & 3396.8 & 3396.8 & 21.85 & 0.000 \\ \text { Residual Error } & 174 & 27053.7 & 155.5 & & \\ \text { Total } & 175 & 30450.5 & & & \end{array} $$
Problem 32
We show an ANOVA table for regression. State the hypotheses of the test, give the F-statistic and the p-value, and state the conclusion of the test. $$ \begin{aligned} &\text { Response: } Y\\\ &\begin{array}{lrrrrr} & \text { Df } & \text { Sum Sq } & \text { Mean Sq } & \mathrm{F} \text { value } & \operatorname{Pr}(>\mathrm{F}) \\ \text { ModelA } & 1 & 352.97 & 352.97 & 11.01 & 0.001 * * \\ \text { Residuals } & 359 & 11511.22 & 32.06 & & \\ \text { Total } & 360 & 11864.20 & & & \end{array} \end{aligned} $$
Problem 33
We show an ANOVA table for regression. State the hypotheses of the test, give the F-statistic and the p-value, and state the conclusion of the test. $$ \begin{aligned} &\text { Response: }\\\ &\begin{array}{lrrrrr} & \text { Df } & \text { Sum Sq } & \text { Mean Sq } & \text { F value } & \operatorname{Pr}(>\text { F) } \\ \text { ModelB } & 1 & 10.380 & 10.380 & 2.18 & 0.141 \\ \text { Residuals } & 342 & 1630.570 & 4.768 & & \\ \text { Total } & 343 & 1640.951 & & & \end{array} \end{aligned} $$
