From the given information , for group 1(unexpected): \(\displaystyle\overline{{{x}}}_{{1}}={9.9},{s}_{{1}}={4.5},{n}={20},\) for group 2(expected): \(\displaystyle\overline{{{x}}}_{{2}}={7.5},{s}_{{2}}={4.2},\) and \(\displaystyle{n}_{{2}}={20}.\) Null hypothesis \(\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}}\) Alternative hypothesis \(\displaystyle{H}_{{\alpha}}:\mu_{{1}}{>}\mu_{{2}}\)

(A) Sample statistic \(\displaystyle\mu_{{{\left({x}_{{1}}-{x}_{{2}}\right)}}}=\overline{{{x}}}_{{1}}-\overline{{{x}}}_{{2}}={9.9}-{7.5}={2.4}\)

(B) The test statistic \(\displaystyle{t}_{{{c}\alpha{1}}}{\frac{{\overline{{{x}}}_{{1}}-\overline{{{x}}}_{{2}}}}{{\sqrt{{{\frac{{{{S}_{{{1}}}^{{{2}}}}}}{{{n}_{{1}}}}}+{\frac{{{{S}_{{{2}}}^{{{2}}}}}}{{{n}_{{2}}}}}}}}}}={\frac{{{9.9}-{7.5}}}{{{\frac{{{\left({4.5}\right)}^{{2}}}}{{{20}}}}+{\frac{{{\left({4.2}\right)}^{{2}}}}{{{20}}}}}}}={1.7437}\)

Degrees of freedom \(\displaystyle={\left({n}_{{1}}+{n}_{{2}}-{2}\right)}={38}\)

(C) \(\displaystyle{P}-{v}{a}{l}{u}{e}={P}{\left({t}{>}{1.7437}\right)}={0.045},\)

\({P}-{v}{a}{l}{u}{e}={0.045}\) less than the level of significance \(\displaystyle\alpha={0.05},\) we reject the null hypothesis.