Question

Suppose the differences in GPAs of all the students in one university in the two most recent semesters (the GPA in the current semester minus the GPA in the last semester) are normally distributed with a mean of 0.2 and a standard deviation of 0.18. What is the probability that a randomly picked student from this university is having a lower GPA in the current semester than what he/she received in the last semester?

Answer 1

Let, define two random variables Y and W that represents GPA of current semester and GPA of last semester.

Let, define a new random variable X that represent the GPA in set current semester minus GPA in last semester.

i.e, X = (Y – W

The random variable X is normally distributed with mean 0.20 and standard deviation 0.18

$i.e, X=(Y-W)\sim N(0.20,0.18^2)$

We need to find the probability that randomly picked student from this university is having a lower GPA in current semester than what he/she received in last semester.

i.e, P(Y <W)

$=P(Y-W<0)$

$=P(X<0)$ [ since, X=Y-W ]

$=P\left ( \frac{X-0.20}{0.18}<\frac{0-0.20}{0.18} \right )$

$=P(Z<-1.1111111)$ [ Z~N(0,1)]

$=0.1332603$ [ from standard normal table we get]

$\simeq 0.1333$ [ round to 4-decimal place]

The probability is 0.1333

The normal distribution curve is given by

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Normal Distribution: Pr(X<0)=0.1333 24 2.2 N o 1 00 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0 2 0.0 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0