Answer :
We are given that the mean of the differences is
[tex]$$\bar{d} = 193,$$[/tex]
the standard deviation is
[tex]$$s_d = 62.73,$$[/tex]
and the sample size is
[tex]$$n = 10.$$[/tex]
Since this is a paired differences setup, the degrees of freedom are
[tex]$$df = n - 1 = 9.$$[/tex]
For a 98% confidence interval, the appropriate formula is
[tex]$$
\bar{d} \pm t^*\frac{s_d}{\sqrt{n}},
$$[/tex]
where [tex]$t^*$[/tex] is the critical value from the [tex]$t$[/tex]-distribution with 9 degrees of freedom. For a 98% confidence level, it has been determined that
[tex]$$
t^*\approx 2.821.
$$[/tex]
Now, compute the standard error:
[tex]$$
SE = \frac{s_d}{\sqrt{n}} = \frac{62.73}{\sqrt{10}}.
$$[/tex]
The margin of error (ME) is then:
[tex]$$
ME = 2.821 \cdot \frac{62.73}{\sqrt{10}}.
$$[/tex]
Thus, the 98% confidence interval for the mean difference is:
[tex]$$
193 \pm 2.821\left(\frac{62.73}{\sqrt{10}}\right).
$$[/tex]
This corresponds to the interval from
[tex]$$
193 - 2.821\left(\frac{62.73}{\sqrt{10}}\right)
$$[/tex]
to
[tex]$$
193 + 2.821\left(\frac{62.73}{\sqrt{10}}\right).
$$[/tex]
So the correct answer is:
[tex]$$
193 \pm 2.821\left(\frac{62.73}{\sqrt{10}}\right).
$$[/tex]
[tex]$$\bar{d} = 193,$$[/tex]
the standard deviation is
[tex]$$s_d = 62.73,$$[/tex]
and the sample size is
[tex]$$n = 10.$$[/tex]
Since this is a paired differences setup, the degrees of freedom are
[tex]$$df = n - 1 = 9.$$[/tex]
For a 98% confidence interval, the appropriate formula is
[tex]$$
\bar{d} \pm t^*\frac{s_d}{\sqrt{n}},
$$[/tex]
where [tex]$t^*$[/tex] is the critical value from the [tex]$t$[/tex]-distribution with 9 degrees of freedom. For a 98% confidence level, it has been determined that
[tex]$$
t^*\approx 2.821.
$$[/tex]
Now, compute the standard error:
[tex]$$
SE = \frac{s_d}{\sqrt{n}} = \frac{62.73}{\sqrt{10}}.
$$[/tex]
The margin of error (ME) is then:
[tex]$$
ME = 2.821 \cdot \frac{62.73}{\sqrt{10}}.
$$[/tex]
Thus, the 98% confidence interval for the mean difference is:
[tex]$$
193 \pm 2.821\left(\frac{62.73}{\sqrt{10}}\right).
$$[/tex]
This corresponds to the interval from
[tex]$$
193 - 2.821\left(\frac{62.73}{\sqrt{10}}\right)
$$[/tex]
to
[tex]$$
193 + 2.821\left(\frac{62.73}{\sqrt{10}}\right).
$$[/tex]
So the correct answer is:
[tex]$$
193 \pm 2.821\left(\frac{62.73}{\sqrt{10}}\right).
$$[/tex]
