Here is a command that generates density plots of `MPG.highway`

from the Cars93 data. Separate densities are constructed for US and non-US vehicles.

```
qplot(data = Cars93, x = MPG.highway,
fill = Origin, geom = "density", alpha = I(0.5))
```

**(a)** Using the Cars93 data and the `t.test()`

function, run a t-test to see if average `MPG.highway`

is different between US and non-US vehicles. *Interpret the results*

Try doing this both using the formula style input and the `x`

, `y`

style input.

```
# Formula version
mpg.t.test <- t.test(MPG.highway ~ Origin, data = Cars93)
mpg.t.test
```

```
##
## Welch Two Sample t-test
##
## data: MPG.highway by Origin
## t = -1.7545, df = 75.802, p-value = 0.08339
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -4.1489029 0.2627918
## sample estimates:
## mean in group USA mean in group non-USA
## 28.14583 30.08889
```

```
# x, y version
with(Cars93, t.test(x = MPG.highway[Origin == "USA"], y = MPG.highway[Origin == "non-USA"]))
```

```
##
## Welch Two Sample t-test
##
## data: MPG.highway[Origin == "USA"] and MPG.highway[Origin == "non-USA"]
## t = -1.7545, df = 75.802, p-value = 0.08339
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -4.1489029 0.2627918
## sample estimates:
## mean of x mean of y
## 28.14583 30.08889
```

There is no statistically significant difference in highway fuel consumption between US and non-US origin vehicles.

**(b)** What is the confidence interval for the difference?

`mpg.t.test$conf.int`

```
## [1] -4.1489029 0.2627918
## attr(,"conf.level")
## [1] 0.95
```

**(c)** Repeat part (a) using the `wilcox.test()`

function.

`mpg.wilcox.test <- wilcox.test(MPG.highway ~ Origin, data = Cars93)`

```
## Warning in wilcox.test.default(x = c(31L, 28L, 25L, 27L, 25L, 25L, 36L, :
## cannot compute exact p-value with ties
```

`mpg.wilcox.test`

```
##
## Wilcoxon rank sum test with continuity correction
##
## data: MPG.highway by Origin
## W = 910, p-value = 0.1912
## alternative hypothesis: true location shift is not equal to 0
```

**(d)** Are your results for (a) and (c) very different?

The p-value from the t-test is somewhat smaller than that output by wilcox.test. Since the MPG.highway distributions are right-skewed, we might expect some differences between the t-test and wilcoxon test Neither test is statistically significant.

**(a)** Modify the density plot code provided in problem 1 to produce a plot with better axis labels. Also add a title.

```
qplot(data = Cars93, x = MPG.highway,
fill = Origin, geom = "density", alpha = I(0.5),
xlab = "Highway fuel consumption (MPG)",
main = "Highway fuel consumption density plots")
```

**(b)** Does the data look to be normally distributed?

The densities donâ€™t really look normally distributed. They appear right-skewed.

**(c)** Construct qqplots of `MPG.highway`

, one plot for each `Origin`

category. Overlay a line on each plot using with `qqline()`

function.

```
par(mfrow = c(1,2))
# USA cars
with(Cars93, qqnorm(MPG.highway[Origin == "USA"]))
with(Cars93, qqline(MPG.highway, col = "blue"))
# Foreign cars
with(Cars93, qqnorm(MPG.highway[Origin == "non-USA"]))
with(Cars93, qqline(MPG.highway, col = "blue"))
```