Question

the marks in an examination for a Physics paper have normal distribution with mean μ and variance σ2 . 10% of the students obtain more than 75 marks and 20% of them obtained less than 40 marks. find the value of  μ  and σ​

Answer 1

Answer:

The mean of the distribution is about 53.9 and standard deviation is about 16.5 marks.

Step-by-step explanation:

Use the trick of transforming the given random variable one that is standard normal distributed, aka a z-score, then look up the two percentile values in z-score tables to get two equations with two unknowns.

So, let x be the variable describing the marks of a student, and

$z=(x-\mu)/\sigma$

the standardized equivalent of that (mu - mean, sigma - standard deviation).

We are looking for values of mu and sigma. At this point we'd be out of luck, but, wait, we're given two bits of info: the 10% point (aka, 90th percentile) and the 20% point (can be interpreted as 100-20th percentile). For each point we can use z-score tables to look up the corresponding values of z (just search for z tables). I found:

z-score for the 10% point: z_10=1.28

z-score for the 20% point: z_20=-0.84

That gives us two equations:

$z_{10}=1.28=(75-\mu)/\sigma\\z_{20}=-0.84=(40-\mu)\sigma$

and can be solved for mu and sigma (do the work on your end, I am showing my result):

$\mu=53.87\,\,,\,\, \sigma=16.51$

The mean of the distribution is about 53.9 and standard deviation is about 16.5 marks.