Sure, let's go through the problem step by step.
To evaluate the t-statistic using the formula:
[tex]\[
t = \frac{\overline{d} - \mu_d}{\frac{s_d}{\sqrt{n}}}
\][/tex]
we need to plug in the given values:
- [tex]\(\overline{d} = -0.625\)[/tex]
- [tex]\(\mu_d = 0\)[/tex]
- [tex]\(s_d = 2.669\)[/tex]
- [tex]\(n = 8\)[/tex]
First, calculate the denominator, which is [tex]\(\frac{s_d}{\sqrt{n}}\)[/tex].
1. Compute the square root of [tex]\(n\)[/tex]:
[tex]\[
\sqrt{n} = \sqrt{8} \approx 2.828
\][/tex]
2. Divide [tex]\(s_d\)[/tex] by [tex]\(\sqrt{n}\)[/tex]:
[tex]\[
\frac{s_d}{\sqrt{n}} = \frac{2.669}{2.828} \approx 0.943
\][/tex]
Now, calculate the numerator:
[tex]\[
\overline{d} - \mu_d = -0.625 - 0 = -0.625
\][/tex]
Next, divide the numerator by the denominator:
[tex]\[
t = \frac{\overline{d} - \mu_d}{\frac{s_d}{\sqrt{n}}} = \frac{-0.625}{0.943} \approx -0.662
\][/tex]
Thus, the value of [tex]\(t\)[/tex] rounded to three decimal places is:
[tex]\[
t \approx -0.662
\][/tex]
