Answer with explanation:
An Unbiased Dice is Rolled 199 times.
Frequency of outcomes 1,2,3,4,5,6 are=28, 29, 47, 40, 22, 33.
Probability of an Event

→→→To check whether the result are significant or not , we will calculate standard error(e) and then z value
1.
![(e_{1})^2=(P_{1})^2+(P'_{1})^2\\\\(e_{1})^2=[\frac{28}{199}]^2+[\frac{33}{199}]^2\\\\(e_{1})^2=\frac{1873}{39601}\\\\(e_{1})^2=0.0472967\\\\e_{1}=0.217478\\\\z_{1}=\frac{P'_{1}-P_{1}}{e_{1}}\\\\z_{1}=\frac{\frac{33}{199}-\frac{28}{199}}{0.217478}\\\\z_{1}=\frac{5}{43.27}\\\\z_{1}=0.12](https://tex.z-dn.net/?f=%28e_%7B1%7D%29%5E2%3D%28P_%7B1%7D%29%5E2%2B%28P%27_%7B1%7D%29%5E2%5C%5C%5C%5C%28e_%7B1%7D%29%5E2%3D%5B%5Cfrac%7B28%7D%7B199%7D%5D%5E2%2B%5B%5Cfrac%7B33%7D%7B199%7D%5D%5E2%5C%5C%5C%5C%28e_%7B1%7D%29%5E2%3D%5Cfrac%7B1873%7D%7B39601%7D%5C%5C%5C%5C%28e_%7B1%7D%29%5E2%3D0.0472967%5C%5C%5C%5Ce_%7B1%7D%3D0.217478%5C%5C%5C%5Cz_%7B1%7D%3D%5Cfrac%7BP%27_%7B1%7D-P_%7B1%7D%7D%7Be_%7B1%7D%7D%5C%5C%5C%5Cz_%7B1%7D%3D%5Cfrac%7B%5Cfrac%7B33%7D%7B199%7D-%5Cfrac%7B28%7D%7B199%7D%7D%7B0.217478%7D%5C%5C%5C%5Cz_%7B1%7D%3D%5Cfrac%7B5%7D%7B43.27%7D%5C%5C%5C%5Cz_%7B1%7D%3D0.12)
→→If the value of z is between 2 and 3 , then the result will be significant at 5% level of Significance.Here value of z is very less, so the result is not significant.
2.
![(e_{2})^2=(P_{2})^2+(P'_{2})^2\\\\(e_{2})^2=[\frac{29}{199}]^2+[\frac{33}{199}]^2\\\\(e_{2})^2=\frac{1930}{39601}\\\\(e_{2})^2=0.04873\\\\e_{2}=0.2207\\\\z_{2}=\frac{P'_{2}-P_{2}}{e_{2}}\\\\z_{2}=\frac{\frac{33}{199}-\frac{29}{199}}{0.2207}\\\\z_{2}=\frac{4}{43.9193}\\\\z_{2}=0.0911](https://tex.z-dn.net/?f=%28e_%7B2%7D%29%5E2%3D%28P_%7B2%7D%29%5E2%2B%28P%27_%7B2%7D%29%5E2%5C%5C%5C%5C%28e_%7B2%7D%29%5E2%3D%5B%5Cfrac%7B29%7D%7B199%7D%5D%5E2%2B%5B%5Cfrac%7B33%7D%7B199%7D%5D%5E2%5C%5C%5C%5C%28e_%7B2%7D%29%5E2%3D%5Cfrac%7B1930%7D%7B39601%7D%5C%5C%5C%5C%28e_%7B2%7D%29%5E2%3D0.04873%5C%5C%5C%5Ce_%7B2%7D%3D0.2207%5C%5C%5C%5Cz_%7B2%7D%3D%5Cfrac%7BP%27_%7B2%7D-P_%7B2%7D%7D%7Be_%7B2%7D%7D%5C%5C%5C%5Cz_%7B2%7D%3D%5Cfrac%7B%5Cfrac%7B33%7D%7B199%7D-%5Cfrac%7B29%7D%7B199%7D%7D%7B0.2207%7D%5C%5C%5C%5Cz_%7B2%7D%3D%5Cfrac%7B4%7D%7B43.9193%7D%5C%5C%5C%5Cz_%7B2%7D%3D0.0911)
Result is not significant.
3.
![(e_{3})^2=(P_{3})^2+(P'_{3})^2\\\\(e_{3})^2=[\frac{47}{199}]^2+[\frac{33}{199}]^2\\\\(e_{3})^2=\frac{3298}{39601}\\\\(e_{3})^2=0.08328\\\\e_{3}=0.2885\\\\z_{3}=\frac{P_{3}-P'_{3}}{e_{3}}\\\\z_{3}=\frac{\frac{47}{199}-\frac{33}{199}}{0.2885}\\\\z_{3}=\frac{14}{57.4279}\\\\z_{3}=0.24378](https://tex.z-dn.net/?f=%28e_%7B3%7D%29%5E2%3D%28P_%7B3%7D%29%5E2%2B%28P%27_%7B3%7D%29%5E2%5C%5C%5C%5C%28e_%7B3%7D%29%5E2%3D%5B%5Cfrac%7B47%7D%7B199%7D%5D%5E2%2B%5B%5Cfrac%7B33%7D%7B199%7D%5D%5E2%5C%5C%5C%5C%28e_%7B3%7D%29%5E2%3D%5Cfrac%7B3298%7D%7B39601%7D%5C%5C%5C%5C%28e_%7B3%7D%29%5E2%3D0.08328%5C%5C%5C%5Ce_%7B3%7D%3D0.2885%5C%5C%5C%5Cz_%7B3%7D%3D%5Cfrac%7BP_%7B3%7D-P%27_%7B3%7D%7D%7Be_%7B3%7D%7D%5C%5C%5C%5Cz_%7B3%7D%3D%5Cfrac%7B%5Cfrac%7B47%7D%7B199%7D-%5Cfrac%7B33%7D%7B199%7D%7D%7B0.2885%7D%5C%5C%5C%5Cz_%7B3%7D%3D%5Cfrac%7B14%7D%7B57.4279%7D%5C%5C%5C%5Cz_%7B3%7D%3D0.24378)
Result is not significant.
4.
![(e_{4})^2=(P_{4})^2+(P'_{4})^2\\\\(e_{4})^2=[\frac{40}{199}]^2+[\frac{33}{199}]^2\\\\(e_{4})^2=\frac{3298}{39601}\\\\(e_{4})^2=0.06790\\\\e_{4}=0.2605\\\\z_{4}=\frac{P_{4}-P'_{4}}{e_{4}}\\\\z_{4}=\frac{\frac{40}{199}-\frac{33}{199}}{0.2605}\\\\z_{4}=\frac{7}{51.8555}\\\\z_{4}=0.1349](https://tex.z-dn.net/?f=%28e_%7B4%7D%29%5E2%3D%28P_%7B4%7D%29%5E2%2B%28P%27_%7B4%7D%29%5E2%5C%5C%5C%5C%28e_%7B4%7D%29%5E2%3D%5B%5Cfrac%7B40%7D%7B199%7D%5D%5E2%2B%5B%5Cfrac%7B33%7D%7B199%7D%5D%5E2%5C%5C%5C%5C%28e_%7B4%7D%29%5E2%3D%5Cfrac%7B3298%7D%7B39601%7D%5C%5C%5C%5C%28e_%7B4%7D%29%5E2%3D0.06790%5C%5C%5C%5Ce_%7B4%7D%3D0.2605%5C%5C%5C%5Cz_%7B4%7D%3D%5Cfrac%7BP_%7B4%7D-P%27_%7B4%7D%7D%7Be_%7B4%7D%7D%5C%5C%5C%5Cz_%7B4%7D%3D%5Cfrac%7B%5Cfrac%7B40%7D%7B199%7D-%5Cfrac%7B33%7D%7B199%7D%7D%7B0.2605%7D%5C%5C%5C%5Cz_%7B4%7D%3D%5Cfrac%7B7%7D%7B51.8555%7D%5C%5C%5C%5Cz_%7B4%7D%3D0.1349)
Result is not significant.
5.
![(e_{5})^2=(P_{5})^2+(P'_{5})^2\\\\(e_{5})^2=[\frac{22}{199}]^2+[\frac{33}{199}]^2\\\\(e_{5})^2=\frac{1573}{39601}\\\\(e_{5})^2=0.03972\\\\e_{5}=0.1993\\\\z_{5}=\frac{P'_{5}-P_{5}}{e_{5}}\\\\z_{5}=\frac{\frac{33}{199}-\frac{22}{199}}{0.1993}\\\\z_{5}=\frac{11}{39.6610}\\\\z_{5}=0.2773](https://tex.z-dn.net/?f=%28e_%7B5%7D%29%5E2%3D%28P_%7B5%7D%29%5E2%2B%28P%27_%7B5%7D%29%5E2%5C%5C%5C%5C%28e_%7B5%7D%29%5E2%3D%5B%5Cfrac%7B22%7D%7B199%7D%5D%5E2%2B%5B%5Cfrac%7B33%7D%7B199%7D%5D%5E2%5C%5C%5C%5C%28e_%7B5%7D%29%5E2%3D%5Cfrac%7B1573%7D%7B39601%7D%5C%5C%5C%5C%28e_%7B5%7D%29%5E2%3D0.03972%5C%5C%5C%5Ce_%7B5%7D%3D0.1993%5C%5C%5C%5Cz_%7B5%7D%3D%5Cfrac%7BP%27_%7B5%7D-P_%7B5%7D%7D%7Be_%7B5%7D%7D%5C%5C%5C%5Cz_%7B5%7D%3D%5Cfrac%7B%5Cfrac%7B33%7D%7B199%7D-%5Cfrac%7B22%7D%7B199%7D%7D%7B0.1993%7D%5C%5C%5C%5Cz_%7B5%7D%3D%5Cfrac%7B11%7D%7B39.6610%7D%5C%5C%5C%5Cz_%7B5%7D%3D0.2773)
Result is not significant.
6.
![(e_{6})^2=(P_{6})^2+(P'_{6})^2\\\\(e_{6})^2=[\frac{33}{199}]^2+[\frac{33}{199}]^2\\\\(e_{6})^2=\frac{2178}{39601}\\\\(e_{6})^2=0.05499\\\\e_{6}=0.2345\\\\z_{6}=\frac{P'_{6}-P_{6}}{e_{6}}\\\\z_{6}=\frac{\frac{33}{199}-\frac{33}{199}}{0.2345}\\\\z_{6}=\frac{0}{46.6655}\\\\z_{6}=0](https://tex.z-dn.net/?f=%28e_%7B6%7D%29%5E2%3D%28P_%7B6%7D%29%5E2%2B%28P%27_%7B6%7D%29%5E2%5C%5C%5C%5C%28e_%7B6%7D%29%5E2%3D%5B%5Cfrac%7B33%7D%7B199%7D%5D%5E2%2B%5B%5Cfrac%7B33%7D%7B199%7D%5D%5E2%5C%5C%5C%5C%28e_%7B6%7D%29%5E2%3D%5Cfrac%7B2178%7D%7B39601%7D%5C%5C%5C%5C%28e_%7B6%7D%29%5E2%3D0.05499%5C%5C%5C%5Ce_%7B6%7D%3D0.2345%5C%5C%5C%5Cz_%7B6%7D%3D%5Cfrac%7BP%27_%7B6%7D-P_%7B6%7D%7D%7Be_%7B6%7D%7D%5C%5C%5C%5Cz_%7B6%7D%3D%5Cfrac%7B%5Cfrac%7B33%7D%7B199%7D-%5Cfrac%7B33%7D%7B199%7D%7D%7B0.2345%7D%5C%5C%5C%5Cz_%7B6%7D%3D%5Cfrac%7B0%7D%7B46.6655%7D%5C%5C%5C%5Cz_%7B6%7D%3D0)
Result is not significant.
⇒If you will calculate the mean of all six z values, you will obtain that, z value is less than 2.So, we can say that ,outcomes are not equally likely at a 0.05 significance level.
⇒⇒Yes , as Probability of most of the numbers that is, 1,2,3,4,5,6 are different, for a loaded die , it should be equal to approximately equal to 33 for each of the numbers from 1 to 6.So, we can say with certainty that loaded die behaves differently than a fair die.