If $Z$ is a continuous random variable which follows a Gaussian distribution with mean=$0$ and standard deviation=1, then
$$\mathbb{P}(Z \leq a)= \int^a_{-\infty}\frac{\exp \{ -z^2 / 2 \}} {\sqrt{2 \pi} }dz=\Phi (a)$$
where $\Phi(a=-2)=0.02,\;\Phi(a=-1.5)=0.067,\;\Phi(a=-1)=0.16,\Phi(a=-0.5)=0.31,\;\Phi(a=0)=0.50,\;\Phi(a=0.5)=0.69,\Phi(a=1)=0.84,\;\Phi(a=1.5)=0.933, \text{and } \Phi(a=2)=0.98.$
Note that $X=\mu+\sigma Z$ follows Gaussian distribution with mean $\mu$ and standard deviation $\sigma$.
Suppose the score distribution of an exam has a Gaussian distribution with mean $\mu$ and standard deviation $\sigma$. A candidate fails if she/he obtains less than $35\%$ marks. However she/he must
$$\begin{array}{|l|c|} \hline \text{Result} & \text{Percentage of Students} \\\hline \text{Passed with distinction} &2\\ \text{Passed without distinction} &82 \\\hline\end{array}$$
Table1: Result of a particular exam taken by $225$ students.
obtain more that $80\%$ marks in order to pass with distinction. The exam is taken by a group of $225$ students and the results are given in Table 1..
You may use the fact that the sample mean $X$ follows a Gaussian distribution with mean $\mu$ and standard deviation $\sigma/\sqrt{n}$.
Which of the following statements are true?
- $\mu-\sigma=35\;\text{and}\;\mu+2\sigma=82$;
- $\mu-2\sigma=16\;\text{and}\;\mu+\sigma=82$;
- $\mu=53.33\;\text{and}\;\sigma=13.33$;
- $\mu=50\;\text{and}\;\sigma=15$;
