Question

The employees of cartwright manufacturing are awarded efficiency ratings. the distribution of the ratings approximates a normal distribution. the mean is 400, the standard deviation 50. what is the area under the normal curve between 400 and 482?

Answer 1

The area under the normal curve between 400 and 482 is
`0.4495`

Explanation:

Let's start defining the random variable.

`X` : ''Efficiency ratings''

We know that the distribution of
`X` approximates a normal distribution ⇒

`X` ~
`N` (μ,σ)

Where the normal distribution is defined by the parameters μ (mean) and σ (standard deviation) ⇒

We know that the mean is 400 and the standard deviation is 50 ⇒

`X` ~
`N`
`(400,50)`

The area under the normal curve between 400 and 482 represents the probability of the variable (
`X` in this case) to assume values between 400 and 482.

We need to calculate :

`P(400<X<482)`

We can standardized this variable by subtracting the mean and then dividing by the standard deviation.

The new variable (X-μ)/σ is called Z

The distribution of Z is

`Z` ~
`N(0,1)`

The probabilities of Z are in any table on internet.

To calculate
`P(Z\leq a)` we can use Φ(a) where Φ is the cumulative function of Z.

Solving the exercise :

`P(400<X<482)` ⇒

`P((400-400)/(50)<Z<(482-400)/(50))` ⇒

`P(0<Z<1.64)`

We find that
`P(400<X<482)=P(0<Z<1.64)`

Looking for the values of the cumulative function of Z in any table we can write :

`P(0<Z<1.64)=` Φ(1.64) - Φ(0) = 0.9495 - 0.500 = 0.4495

We find that the probability is 0.4495 and therefore the area under the normal curve (of X) between 400 and 482 is 0.4495

Answer 2

A suitable probability calculator will tell you that area is about 0.44950.