Question: Let \(\mathrm{X}_{1}, \mathrm{X}_{2}, \ldots, \mathrm{X}_{16}\), be a sequence of 16 random sample from a normally distributed population with mean 20 and variance 16 , and \(Y_{1}, Y_{2}, \ldots ., Y_{9}\), be a sequence of 9 random sample from a normally distributed population with mean 10 and variance 25. Defines:

\[\begin{array}{ll} \bar{X}=\frac{\sum_{i=1}^{16} X_{i}}{16} & S_{x}^{2}=\frac{\sum_{i=1}^{16}\left(X_{i}-\bar{X}\right)^{2}}{15} \\ \bar{Y}=\frac{\sum_{j=1}^{9} Y_{j}}{9} & S_{Y}^{2}=\frac{\sum_{j=1}^{9}\left(Y_{j}-\bar{Y}\right)^{2}}{8} \end{array}\]

1. What is the value of \(\mathrm{P}\{\overline{\mathrm{X}}>18.75\}\) ?
2. What is the value of \(\mathrm{P}\{\overline{\mathrm{Y}}<10-0.62 \mathrm{Sy}\} ?\)
3. What is the value of \(\mathrm{P}\left\{\mathrm{S}_{\mathrm{x}}^{2}>9.1168\right\} ?\)
4. What is the value of \(\mathrm{P}\left\{\mathrm{S}_{\mathrm{Y}}^{2}<6.25 \mathrm{~S}_{\mathrm{X}}^{2}\right\} ?\)
5. What is the value of \(\mathrm{P}\{|4 \overline{\mathrm{X}}-80|<1.341 \mathrm{Sx}\} ?\)
6. What is the value of \(\mathrm{P}\left\{\mathrm{X}_{1}+\mathrm{X}_{2}>6.314\left|\mathrm{X}_{1}-\mathrm{X}_{2}\right|+40\right\} ?\)
7. What is the value of \(\mathrm{P}\left\{\sum_{i=1}^{4}\left(\mathrm{X}_{i}-20\right)^{2}+0.64 \sum_{j=1}^{3}\left(\mathrm{Y}_{j}-10\right)^{2}<192.32\right\} ?\)
8. What is the value of \(\mathrm{P}\left\{\mathrm{X}_{2}-\mathrm{Y}_{2}-\mathrm{Y}_{4}>\sqrt{66}\right\}\)
9. What is the value of \(\mathrm{P}\left\{\sum_{i=1}^{5}\left(\mathrm{X}_{i}-20\right)^{2}>7.488 \sum_{j=1}^{4}\left(\mathrm{Y}_{j}-10\right)^{2}\right\} ?\)
10. What is the value of \(\mathrm{P}\{\overline{\mathrm{X}}-\overline{\mathrm{Y}}>10+\sqrt{1.36}\} ?\)

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