Now let us consider the case of testing two means from dependent samples also sometimes known as matched pairs data. Matched pairs occur when the two samples are somehow related to one another. For example arm$-$span and height of the same person are typically considered to be related or dependent on each other. Another example is if we are trying to test to see if men react differently then women to a medicine we want to match on as many characteristics as possible so that any effect by gender will stand out. Twin studies are another example of matched pairs.

The first step in any matched pairs problem is to construct a new data set which is just the differences between each matched pair on the characteristic of interest. For example if wet pavement versus dry pavement effects breaking distance, we would construct a data set that would be made up of either $($Wet breaking distances $-$Dry breaking distances$)$ or $($Dry breaking distances$-$Wet breaking distances$).$ The order impacts what our alternative hypothesis is$.$ The points in the new data set are labeled d$1,$ d$2,$ d$3,,$ dn were n is the sample size. Then we find the mean and standard deviation of the new data set and label them $$d and sd$.$ The null hypothesis is that the mean difference is $0.$ The test statistic is given below:

$$sdn$=$d$$sdn

t$0=$d $0$

which follows Student$$s t$-$distribution with n$-1$ degrees of freedom. The values of $$d and sd are the mean and standard deviation of the differenced data. The data points we are looking at are of the form di $=$ Xi $$ Yi where Xi and Yi are points from the original samples.

$5.$ A Hartwick pre$-$Engineering student wishes to determine if on average there is a difference in braking times for when the pavement is covered in wet leaves and when the pavement is simply wet. To help reduce bias, the student uses the driver every time and uses $10$ different models of cars. After creating the differenced data the sample mean of the differenced data is $$d $=32\mathrm{.}7$ and the sample standard deviation of the differenced data is sd $=3\mathrm{.}7671513$

$($a$)$ How many samples do we have.

$($b$)$ If there are two samples are they independent of dependent?

$($c$)$ What is the parameter type?

$($d$)$ What is the null hypothesis? Explain your reasoning

$($e$)$ What is the alternative hypothesis? Explain your reasoning.

$($f$)$ Is your test two$-$tailed, left$-$tailed or right$-$tailed?

$($g$)$ What is the level of significance $\backslash$alpha $?$

$($h$)$ What is the test statistic?

$($i$)$ Do we reject the null or fail to reject the null.

$($j$)$ Interpret your result.