Step 1

From the provided information,

\(\displaystyle{n}_{{1}}={41}\)

\(\displaystyle\overline{{x}}_{{1}}={33}\)

\(\displaystyle{n}_{{2}}={48}\)

\(\displaystyle\overline{{x}}_{{2}}={32}\)

\(\displaystyle{s}_{{1}}={9}\)

\(\displaystyle{s}_{{2}}={10}\)

The confidence level = 98%

Level of significance \(\displaystyle{\left(\alpha\right)}={1}-{0.98}={0.02}\)

The degree of freedom \(\displaystyle={n}_{{1}}+{n}_{{2}}-{2}={41}+{48}-{2}={87}\)

The critical value of t at 87 degree of freedom with 0.02 level of significance from the t value table is 2.37.

Step 2

The pooled standard deviation can be obtained as:

\(\displaystyle{s}_{{p}}=\sqrt{{\frac{{{\left({n}_{{1}}-{1}\right)}{{s}_{{1}}^{{2}}}+{\left({n}_{{2}}-{1}\right)}{{s}_{{2}}^{{2}}}}}{{{n}_{{1}}+{n}_{{2}}-{2}}}}}\)

\(\displaystyle=\sqrt{{\frac{{{\left({41}-{1}\right)}{\left({9}\right)}^{{2}}+{\left({48}-{1}\right)}{\left({10}\right)}^{{2}}}}{{{41}+{48}-{2}}}}}\)

=9.553

The required margin of error can be obtained as:

\(\displaystyle{E}={t}{\left({s}_{{p}}\sqrt{{\frac{{1}}{{{n}_{{1}}}}+\frac{{1}}{{{n}_{{2}}}}}}\right)}\)

\(\displaystyle={\left({2.37}\right)}{\left({9.553}\sqrt{{\frac{{1}}{{41}}+\frac{{1}}{{48}}}}\right)}\)

\(\displaystyle\approx{4.81}\)

Thus, the margin of error is 4.81.

Step 3

The required 98% confidence interval for \(\displaystyle\mu_{{1}}-\mu_{{2}}\) can be obtained as:

\(\displaystyle{C}{I}={\left(\overline{{x}}_{{1}}-\overline{{x}}_{{1}}\right)}\pm{E}\)

\(\displaystyle={\left({33}-{32}\right)}\pm{4.81}\)

=1+-4.81

=(-3.81,5.81)

Thus, the lower limit of confidence interval is -3.81 and the upper limit is 5.81.

From the provided information,

\(\displaystyle{n}_{{1}}={41}\)

\(\displaystyle\overline{{x}}_{{1}}={33}\)

\(\displaystyle{n}_{{2}}={48}\)

\(\displaystyle\overline{{x}}_{{2}}={32}\)

\(\displaystyle{s}_{{1}}={9}\)

\(\displaystyle{s}_{{2}}={10}\)

The confidence level = 98%

Level of significance \(\displaystyle{\left(\alpha\right)}={1}-{0.98}={0.02}\)

The degree of freedom \(\displaystyle={n}_{{1}}+{n}_{{2}}-{2}={41}+{48}-{2}={87}\)

The critical value of t at 87 degree of freedom with 0.02 level of significance from the t value table is 2.37.

Step 2

The pooled standard deviation can be obtained as:

\(\displaystyle{s}_{{p}}=\sqrt{{\frac{{{\left({n}_{{1}}-{1}\right)}{{s}_{{1}}^{{2}}}+{\left({n}_{{2}}-{1}\right)}{{s}_{{2}}^{{2}}}}}{{{n}_{{1}}+{n}_{{2}}-{2}}}}}\)

\(\displaystyle=\sqrt{{\frac{{{\left({41}-{1}\right)}{\left({9}\right)}^{{2}}+{\left({48}-{1}\right)}{\left({10}\right)}^{{2}}}}{{{41}+{48}-{2}}}}}\)

=9.553

The required margin of error can be obtained as:

\(\displaystyle{E}={t}{\left({s}_{{p}}\sqrt{{\frac{{1}}{{{n}_{{1}}}}+\frac{{1}}{{{n}_{{2}}}}}}\right)}\)

\(\displaystyle={\left({2.37}\right)}{\left({9.553}\sqrt{{\frac{{1}}{{41}}+\frac{{1}}{{48}}}}\right)}\)

\(\displaystyle\approx{4.81}\)

Thus, the margin of error is 4.81.

Step 3

The required 98% confidence interval for \(\displaystyle\mu_{{1}}-\mu_{{2}}\) can be obtained as:

\(\displaystyle{C}{I}={\left(\overline{{x}}_{{1}}-\overline{{x}}_{{1}}\right)}\pm{E}\)

\(\displaystyle={\left({33}-{32}\right)}\pm{4.81}\)

=1+-4.81

=(-3.81,5.81)

Thus, the lower limit of confidence interval is -3.81 and the upper limit is 5.81.