Question

for a normal distribution with u=40 with õ = 4 whatbis the probability of selecting a score greater than x = 44

Answer 1

Let X be a random variable that is distributed as a normal distribution with mean 40 and standard deviation = 4. We want to calculate following probability

`P(X>44)`

To do so, we will use a standard normal distribution, which is a normal distribution with mean 0 and standard deviation 1. So, we will transform the variable X in a variable Z that has a standard normal distribution. To do so we subtract the mean of X to X and then divide it by its standard deviation. That is, define

`Z=\frac{X\text{ -40}}{4}`

So, this variable Z has a standard normal distribution with mean 0 and standard deviation of 1.

So, we want to translate this probability

`P(X>44)\text{ }`

to a probabilty using the variable Z. So we if we start with this inequality

`X>44`

if we subtract 40 on both sides, we get

`X\text{ -40 > 44-40=4}`

Now, if we divide both sides by 4, we get

`\frac{X\text{ -40}}{4}>(4)/(4)=1`

So the initial inequality is the same as the following

`Z>1`

So, we have the following equivalence

`P(X>44)=P(Z>1)`

Using the properties of probability, we have that

`P(Z>1)\text{ = 1 - P(Z<=1) }=1\text{ - P(Z<1)}`

Using a table for the left side area of a standard normal distribution, we have that

`P(Z<1)\text{ =0.84134}`

So we have

`P(X>44)\text{ = P(Z>1) = 1-0.84134 = 0.15866}`

So, the probability of selecting a score greater than 44 is 0.15866