A national insurance organizastion wanted to study the consumption pattern of cigarettes in all 50 states and the Distric of Columbia. The variables chosen for the study are given in the table below.

The data from 1970 is cigarette.txt, where the states are given in alphabetical order. The following regression output was obtained using R.

(a) State a multiple linear regression model relating Y to the six variables. State the estimated regression model as well.

(b) Suppose that the researchers did some further analysis and the R output results are as follow:

fit0=lm(Y~1,data=ciga2)

anova(fit0,fit)

Explain the purpose of the statistical analysis and the results in details.

(c) Suppose that they were also interested in a problem of testing whether HS = 2Black and 10Income = Black. Specify the null and alternative hypotheses and the test statistics, and write down your R code for doing the test. (d) The following results show one iterate in the middle of backward elimination using Ftests at 5% significance level. For the iterate, please state the variable dropped out, its p-value and the corresponding null and alternative hypotheses. Note that drop1 function performs F-test regarding deletions of the each covariates from the given model.

Variable Definition Age Median age of a person living in a state HS Percentage of people over 25 years of age in a state who had completed high school Per capita personal income for a state income in dollars) Income Black Percentage of blacks living in a state Female Percentage of females living in a state Price Weighted average price in cents) of a pack of cigarettes in a state Sales Number of packs of cigarettes sold in a state on a per capita basis ciga = read.table("cigarette.txt", header=T) str(ciga) $ Age ## 'data.frame': 51 obs. of 8 variables: ## State: Factor w/ 51 levels "AK", "AL", "AR",..: 2 1 4 3 5 6 7 9 8 10 ## : num 27 22.9 26.3 29.1 28.1 26.2 29.1 26.8 28.4 32.3 ... ## $ HS : num 41.3 66.7 58.1 39.9 62.6 63.9 56 54.6 55.2 52.6 ## $ Income: num 2948 4644 3665 2878 4493 ## $ Black : num 26.2 3 3 18.3 7 3 6 14.3 71.1 15.3 ... ## $ Female: num 51.7 45.7 50.8 51.5 50.8 50.7 51.5 51.3 53.5 51.8 ## $ Price : num 42.7 41.8 38.5 38.8 39.7 31.1 45.5 41.3 32.6 43.8 ## $ Sales : num 89.8 121.3 115.2 100.3 123 ... head(ciga) ## ## 1 ## 2 ## 3 ## 4 ## 5 ## 6 State Age HS Income Black Female Price Sales AL 27.0 41.3 2948 26.2 51.7 42.7 89.8 AK 22.9 66.7 4644 3.0 45.7 41.8 121.3 AZ 26.3 58.1 3665 3.0 50.8 38.5 115.2 AR 29.1 39.9 2878 18.3 51.5 38.8 100.3 CA 28.1 62.6 4493 7.0 50.8 39.7 123.0 CO 26.2 63.9 3855 3.0 50.7 31.1 124.8 Y=log(ciga$Sales) ciga2 = data.frame (Y=log(ciga$Sales), ciga[,-c(1,8)]) fit Im(Y., data=ciga2) fit . ## ## Call: ## lm(formula = Y data = ciga2) ## ## Coefficients: ## (Intercept) Age HS ## 4.8217962 0.0388336 -0.0034839 ## Female Price ## -0.0129806 -0.0243805 Income 0.0001702 Black 0.0018311 ## Single term deletions ## ## Model: ## Y Age + HS + Income + Black + Price ## Df Sum of Sq RSS AIC F value Pr(>F) ## 1.5939 -164.75 1 0.15583 1.7498 -161.99 4.3995 0.0415976 * ## HS 1 0.01508 1.6090 -166.27 0.4256 0.5174615 ## Income 1 0.25486 1.8488 -159.18 7.1953 0.0101862 * ## Black 1 0.00711 1.6010 -166.52 0.2007 0.6563285 ## Price 1 0.44233 2.0363 -154.26 12.4878 0.0009608 *** ## ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05.' 0.1 ' ## Age > 1 Variable Definition Age Median age of a person living in a state HS Percentage of people over 25 years of age in a state who had completed high school Per capita personal income for a state income in dollars) Income Black Percentage of blacks living in a state Female Percentage of females living in a state Price Weighted average price in cents) of a pack of cigarettes in a state Sales Number of packs of cigarettes sold in a state on a per capita basis ciga = read.table("cigarette.txt", header=T) str(ciga) $ Age ## 'data.frame': 51 obs. of 8 variables: ## State: Factor w/ 51 levels "AK", "AL", "AR",..: 2 1 4 3 5 6 7 9 8 10 ## : num 27 22.9 26.3 29.1 28.1 26.2 29.1 26.8 28.4 32.3 ... ## $ HS : num 41.3 66.7 58.1 39.9 62.6 63.9 56 54.6 55.2 52.6 ## $ Income: num 2948 4644 3665 2878 4493 ## $ Black : num 26.2 3 3 18.3 7 3 6 14.3 71.1 15.3 ... ## $ Female: num 51.7 45.7 50.8 51.5 50.8 50.7 51.5 51.3 53.5 51.8 ## $ Price : num 42.7 41.8 38.5 38.8 39.7 31.1 45.5 41.3 32.6 43.8 ## $ Sales : num 89.8 121.3 115.2 100.3 123 ... head(ciga) ## ## 1 ## 2 ## 3 ## 4 ## 5 ## 6 State Age HS Income Black Female Price Sales AL 27.0 41.3 2948 26.2 51.7 42.7 89.8 AK 22.9 66.7 4644 3.0 45.7 41.8 121.3 AZ 26.3 58.1 3665 3.0 50.8 38.5 115.2 AR 29.1 39.9 2878 18.3 51.5 38.8 100.3 CA 28.1 62.6 4493 7.0 50.8 39.7 123.0 CO 26.2 63.9 3855 3.0 50.7 31.1 124.8 Y=log(ciga$Sales) ciga2 = data.frame (Y=log(ciga$Sales), ciga[,-c(1,8)]) fit Im(Y., data=ciga2) fit . ## ## Call: ## lm(formula = Y data = ciga2) ## ## Coefficients: ## (Intercept) Age HS ## 4.8217962 0.0388336 -0.0034839 ## Female Price ## -0.0129806 -0.0243805 Income 0.0001702 Black 0.0018311 ## Single term deletions ## ## Model: ## Y Age + HS + Income + Black + Price ## Df Sum of Sq RSS AIC F value Pr(>F) ## 1.5939 -164.75 1 0.15583 1.7498 -161.99 4.3995 0.0415976 * ## HS 1 0.01508 1.6090 -166.27 0.4256 0.5174615 ## Income 1 0.25486 1.8488 -159.18 7.1953 0.0101862 * ## Black 1 0.00711 1.6010 -166.52 0.2007 0.6563285 ## Price 1 0.44233 2.0363 -154.26 12.4878 0.0009608 *** ## ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05.' 0.1 ' ## Age > 1