The built-in data set \"mtcars\" compares 11 aspects of automobile design for 32

Question

The built-in data set "mtcars" compares 11 aspects of automobile design for 32 different 1974 model automobiles. We will be looking at the wt column of mtcars. Assume that the 32 cars are a random sample of all 1974 automobiles. We would like to estimate the true mean value, , of the wt (weight in 1000's of pounds) of cars in 1974. The unknown variance of wt is 2. Using R define the vector x by x<-mtcars$wt. A screen print of the data follows.

[1] 2.620 2.875 2.320 3.215 3.440 3.460 3.570 3.190 3.150 3.440 3.440 4.070 3.730
[14] 3.780 5.250 5.424 5.345 2.200 1.615 1.835 2.465 3.520 3.435 3.840 3.845 1.935
[27] 2.140 1.513 3.170 2.770 3.570 2.780

e) Assuming normality of car weights, calculate the maximum likelihood estimate of  ?  

f) Calculate the 65th percentile of x using R.  

g) Calculate a 10% trimmed mean for x using R.  

h) Since the sample size is >30 we can create a confidence interval for using a normal critical value. If we want the confidence interval to be at the 96% level and we use a normal critical value, then what critical value should we use?  

i) Calculate a 96% confidence interval(using a normal critical value) for .

j) How long is the 96% confidence interval just created in part i?

Explanation / Answer

## Program

X=c(2.620, 2.875, 2.320, 3.215, 3.440, 3.460, 3.570, 3.190, 3.150, 3.440,
3.440, 4.070, 3.730, 3.780, 5.250, 5.424, 5.345, 2.200, 1.615, 1.835,
2.465, 3.520, 3.435, 3.840, 3.845, 1.935, 2.140, 1.513, 3.170, 2.770,
3.570, 2.780)

## e)
mu=mean(X)
mu

## f)
quantile(X, 0.65)

# g)
mean(X, trim=0.10)

## h)
Z=qnorm(0.98, mean = 0, sd = 1)
Z

## i) i) Calculate a 96% confidence interval(using a normal critical value) for .
s=sd(X)
n=length(X)

Lower_bound=mu-Z*s/sqrt(n)
Lower_bound

Upper_bound=mu+Z*s/sqrt(n)
Upper_bound

## j) How long is the 96% confidence interval just created in part i?
Long=Upper_bound-Lower_bound
Long

## End Command

## Rund the command

> X=c(2.620, 2.875, 2.320, 3.215, 3.440, 3.460, 3.570, 3.190, 3.150, 3.440,
+ 3.440, 4.070, 3.730, 3.780, 5.250, 5.424, 5.345, 2.200, 1.615, 1.835,
+ 2.465, 3.520, 3.435, 3.840, 3.845, 1.935, 2.140, 1.513, 3.170, 2.770,
+ 3.570, 2.780)
>
> ## e)
> mu=mean(X)
> mu
[1] 3.21725
>
>
> ## f)
> quantile(X, 0.65)
65%
3.469
>
> # g)
> mean(X, trim=0.10)
[1] 3.152692
>
> ## h)
> Z=qnorm(0.98, mean = 0, sd = 1)
> Z
[1] 2.053749
>
> ## i) i) Calculate a 96% confidence interval(using a normal critical value) for .
> s=sd(X)
> n=length(X)
>
> Lower_bound=mu-Z*s/sqrt(n)
> Lower_bound
[1] 2.862016
>
> Upper_bound=mu+Z*s/sqrt(n)
> Upper_bound
[1] 3.572484
>
> ## j) How long is the 96% confidence interval just created in part i?
> Long=Upper_bound-Lower_bound
> Long
[1] 0.7104676

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