Question: Assume that the length X in minutes of a regular season NBA game is normally Distributed with \[X\tilde{\ }N({{\mu }_{X}},{{\sigma }_{X}}^{2})\]; and similarly the Y in minutes of an NBA playoff game is also normally distributed with \[Y\tilde{\ }N({{\mu }_{Y}},{{\sigma }_{Y}}^{2})\]. Six recent regular season games took 98, 100, 95, 120, 103, and 114 minutes \[(\sum{{{x}_{i}}=630}\] and \[{{s}_{X}}^{2}=100)\]. Four very recent playoff games took 98, 122, 106, and 118 minutes \[(\sum{{{y}_{i}}=444}\] and \[{{s}_{Y}}^{2}=121.3)\]

a.) Assuming \[{{\sigma }_{X}}^{2}\] and \[{{\sigma }_{Y}}^{2}\] are unknown but the same, test \[{{H}_{0}}:{{\mu }_{X}}={{\mu }_{Y}}\] versus \[{{H}_{1}}:{{\mu }_{X}}\ne {{\mu }_{Y}}\] at the \[\alpha =0.10\] level of significance.

b.) Test, with \[\alpha =0.10\], the assumption that \[{{\sigma }_{X}}^{2}\] = \[{{\sigma }_{Y}}^{2}\]