In: Math

# As part of a study on transportation safety, the U.S. Department of Transportation collected data on...

As part of a study on transportation safety, the U.S. Department of Transportation collected data on the number of fatal accidents per 1000 licenses and the percentage of licensed drivers under the age of 21 in a sample of 42 cities. Data collected over a one-year period follow. These data are contained in the file named “Safety.csv”.

1- Find the sample mean and standard deviation for each variable. Round your answers to the nearest thousandth.

2- Use the function lm() in R to run a simple linear regression model on the data provided. Use the function summary() in R to generate the regression output. Use the function aov() in R to generate the corresponding ANOVA table. You ought to be able to determine which is the dependent variable and which is the independent variable in this SLR model.

Please copy your R code and the result and paste them here.

3- Write down the estimated regression function below and provide a practical interpretation of the coefficient of the independent variable.

4- Please find a 95% confidence interval for the coefficient of the independent variable and provide a practical interpretation of this interval.

5- At the 5% level of significance, is there a significant relationship between the two variables? Why or why not?

6- What is the value of the coefficient of determination for this simple linear regression model? Provide a brief interpretation of this value.

7- Use the information from the ANOVA table to compute the standard error of estimate, a.k,a, residual standard error. This value must match the residual standard error in the regression summary.

8- What is the point estimate of the expected number of fatal accidents per 1000 licenses if there are 10% drivers under age in a city?

9- Suppose we want to develop a 95% confidence interval for the average number of fatal accidents per 1000 licenses for all the cities with 10% of drivers under age 21. What is the estimate of the standard deviation for this confidence interval?

10-Suppose we want to develop a 95% confidence interval for the average number of fatal accidents per 1000 licenses for all the cities with 10% of drivers under age 21. Compute the t value and the margin of error needed for this confidence interval.

Please copy your R code and the result and paste them here.

11-Provide a 95% confidence interval for the average number of fatal accidents per 1000 licenses for all the cities with 10% of drivers under age 21 and a practical interpretation to this confidence interval.

12- Suppose we want to develop a 95% prediction interval for the number of fatal accidents per 1000 licenses for a city with 10% of drivers under age 21. What is the estimate of the standard deviation for this prediction interval?

13- Suppose we want to develop a 95% prediction interval for the number of fatal accidents per 1000 licenses for a city with 10% of drivers under age 21. Compute the margin of error needed for this prediction interval.

14- Provide a 95% prediction interval for the number of fatal accidents per 1000 licenses for a city with 10% of drivers under age 21 and a practical interpretation to this prediction interval.

 Percent Under 21 Fatal Accidents per 1000 13 2.962 12 0.708 8 0.885 12 1.652 11 2.091 17 2.627 18 3.83 8 0.368 13 1.142 8 0.645 9 1.028 16 2.801 12 1.405 9 1.433 10 0.039 9 0.338 11 1.849 12 2.246 14 2.855 14 2.352 11 1.294 17 4.1 8 2.19 16 3.623 15 2.623 9 0.835 8 0.82 14 2.89 8 1.267 15 3.224 10 1.014 10 0.493 14 1.443 18 3.614 10 1.926 14 1.643 16 2.943 12 1.913 15 2.814 13 2.634 9 0.926 17 3.256

Ps: I do appreciate your help But please do not simply copy and paste irrelevant answer, Thanks

## Solutions

##### Expert Solution

The R code for Q 1-4

round(mean(data[,1]),3)

# 12.262

round(mean(data[,2]),3)

# 1.922

round(sd(data[,1]),3)

# 3.132

round(sd(data[,2]),3)

# 1.071

names(data)

model1=lm(Accidents~Age,data=data)

summary(model1)

#Call:

# lm(formula = Accidents ~ Age, data = data)

#Residuals:

# Min 1Q Median 3Q Max

#-1.23412 -0.26441 0.00772 0.44362 1.49099

#Coefficients:

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

#(Intercept) -1.59741 0.37167 -4.298 0.000107 ***

# Age 0.28705 0.02939 9.767 3.79e-12 ***

# ---

# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#Residual standard error: 0.5894 on 40 degrees of freedom

#Multiple R-squared: 0.7046, Adjusted R-squared: 0.6972

#F-statistic: 95.4 on 1 and 40 DF, p-value: 3.794e-12

aov(model1)

#Call:

# aov(formula = model1)

#Terms:

# Age Residuals

#Sum of Squares 33.13442 13.89335

#Deg. of Freedom 1 40

#Residual standard error: 0.5893503

#Estimated effects may be unbalanced

confint(model1,"Age",level = 0.95)

# 2.5 % 97.5 %

#Age 0.2276542 0.3464521

The estimated regression equation is y = -1.59741 + 0.28705 * (% under 21). The coefficient .28705 means that if the percentage of under 21 licensed drivers increases by 1 then the predicetd value of fatal accidents per 1000 increases by .28705.

The confidence interval found above does not contain 0, hence we can be 95% confident that the age variable is significant

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