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
$$\begin{array}{|l|c|} \hline \text{Result} & \text{Percentage of Students} \\\hline \text{Passed with distinction} &2\\ \text{Passed without distinction} &82 \\\hline\end{array}$$
$$\text{Table1: Result of a particular exam taken by $225$ students.}$$
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?
- The probability that the average score of the group of $225$ students is less than $48$ is $2\%$
- The probability that the average score of the group of $225$ students is less than $42.5$ is $6.7\%$
- The probability that the average score of the group of $225$ students is greater than $49$ is more than $50\%$
- The probability that the average score of the group of $225$ students is greater than $57.5$ is more than $16\%$
