To construct a 95% confidence interval for [tex]\( \mu_d \)[/tex], the population mean difference in tread wear between the two brands, we will follow these steps:
1. Calculate the mean of the sample differences:
[tex]\[
\bar{d} = \text{mean of the differences}
\][/tex]
Given the sample differences are:
[tex]\[
[-0.3, 0.34, -0.9, 0.03, -1.03, 0.12, -0.12, -0.15, -0.28, -0.02, -0.25, -0.74, -0.32]
\][/tex]
We obtain:
[tex]\[
\bar{d} = -0.278
\][/tex]
2. Calculate the standard deviation of the sample differences:
[tex]\[
s_d = \text{standard deviation of the differences}
\][/tex]
We obtain:
[tex]\[
s_d = 0.399
\][/tex]
3. Calculate the standard error of the mean difference:
[tex]\[
SE = \frac{s_d}{\sqrt{n}}
\][/tex]
where [tex]\( n \)[/tex] is the number of differences. For our sample:
[tex]\[
n = 13
\][/tex]
We obtain:
[tex]\[
SE = \frac{0.399}{\sqrt{13}} \approx 0.111
\][/tex]
4. Determine the critical value:
For a 95% confidence interval and [tex]\( n - 1 = 12 \)[/tex] degrees of freedom, the critical value ([tex]\( t \)[/tex] value) from the t-distribution is:
[tex]\[
t_{\alpha/2} \approx 2.179
\][/tex]
5. Calculate the margin of error:
[tex]\[
ME = t_{\alpha/2} \times SE
\][/tex]
We obtain:
[tex]\[
ME = 2.179 \times 0.111 \approx 0.241
\][/tex]
6. Calculate the confidence interval:
[tex]\[
\text{Lower limit} = \bar{d} - ME
\][/tex]
[tex]\[
\text{Upper limit} = \bar{d} + ME
\][/tex]
We obtain:
[tex]\[
\text{Lower limit} = -0.278 - 0.241 \approx -0.520
\][/tex]
[tex]\[
\text{Upper limit} = -0.278 + 0.241 \approx -0.037
\][/tex]
Therefore, the 95% confidence interval for the population mean difference in tread wear between the two brands is approximately:
[tex]\[
\text{Lower limit}: -0.52
\][/tex]
[tex]\[
\text{Upper limit}: -0.04
\][/tex]
