An insurance company works with two auto repair shops and wants to compare their prices. Company officials randomly select 12 damaged cars, take each car to both shops, and ask how much the repairs will cost. For each car, they subtract the cost at the first shop from the cost at the second shop. Which of the following would be the most appropriate pair of hypotheses for them to use?

A) H0:μ1=μ2

B) H0:μ1≠μ2

C) H0:μd=0 and Ha:μd≠

D) H0:μd≠0 and Ha:μd=0

A paired t-test is used to compare two population means where you have two samples in which observations in one sample can be paired with observations in the other sample. For example if we have Before-and-after observations (This problem) we can use it.

Let put some notation

x=value for the first shop , y = value for the second shop

The system of hypothesis for this case are:

Null hypothesis: [tex]\mu_y- \mu_x = 0[/tex]

Alternative hypothesis: [tex]\mu_y -\mu_x \neq 0[/tex]

Or equivalently :

Null hypothesis: [tex]\mu_d = 0[/tex]

Alternative hypothesis: [tex]\mu_d \neq 0[/tex]

Since we define the difference [tex]d_i=y_i-x_i[/tex] and we obtain this:

The second step is calculate the mean difference

[tex]\bar d= \frac{\sum_{i=1}^n d_i}{n}[/tex]

The third step would be calculate the standard deviation for the differences, and we got:

[tex]s_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n-1}[/tex]

The 4 step is calculate the statistic given by :

[tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=t_{calculated}[/tex]

The next step is calculate the degrees of freedom given by:

[tex]df=n-1=12-1=11[/tex]

Now we can calculate the p value, since we have a two tailed test the p value is given by:

[tex]p_v =2*P(t_{(11)}>|t_{calculated}|) [/tex]

C) H0:μd=0 and Ha:μd≠ 0