A researcher uses data from 162 U.S. cities in 1900 to investigate the link between infant mortality and toxic lead in drinking water. Her data set contains the following variables: infrate (infant mortality rate), lead (a dummy variable equal to 1 if the city had lead pipes and 0 otherwise) and ph (water pH, where pH is a measure of acidity: in particular, lower pH denotes higher acidity). The following is her R output.

• lead_mortality<-read.csv("lead_mortality.csv")
• modi<-Im(formula=infrate~lead+ph+lead:ph, data=lead_mortality)
• summary (mod1)

Call:

Im(formula = infrate ~ lead + ph + lead:ph, data = lead_mortality)

Residuals:

Min

10

Median

30

Max

-0.27645 -0.08816 0.00134 0.07668 0.36350

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.81337

0.17696

4.596 8.75e-06

lead

0.59436

0.22733

2.615

0.0098 **

ph

-0.06164 0.02456 -2.510

0.0131

lead: ph

-0.07394 0.03113

-2.375

0.0187 *

Signif. codes: 0 ****' 0.001 ***' 0.01 **' 0.05 .' 0.1 * ' 1

Residual standard error: 0.1279 on 158 degrees of freedom

Multiple R-squared: 0.2694,

Adjusted R-squared: 0.2556

F-statistic: 19.42 on 3 and 158 DF, -value: 9.006-11

> bptest (mod1) studentized Breusch-Pagan test

data: mod1

BP = 10.013, df = 3, p-value = 0.01845

> ( coeftest (modi, vcov = VCOVHC) )

t test of coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.813374

0.152818 5.3225 3.457e-07 #**

lead

0.594364

0.219825 2.7038 0.007606 **

ph

-0.061640 0.021637 -2.8489 0.004972 **

lead: ph

-0.073944 0.029850 -2.4772 0.014295 *

Signif. codes: 0 **#*' 0.001 **#' 0.01 **' 0.05 '.' 0

(a) Does her test detect heteroskedasticity at 5% level of significance?

(b) Consider two linear regressions: a regression of infrate on ph for cities with lead pipes and a regression of infrate on ph for cities without lead pipes

(i) Are the intercepts of the two regressions equal at 5% level of significance? What is the corresponding t-statistic?

(ii) Are the slopes of the two regressions equal at 5% level of significance? What is the corresponding t-statistic?

(c) Find the predicted difference in infrate between a city with lead pipes and a city without lead pipes if the water in both cities has a pH of 7.