Answer:
The area under the normal curve between 400 and 482 is [tex]0.4495[/tex]
Step-by-step explanation:
Let's start defining the random variable.
[tex]X[/tex] : ''Efficiency ratings''
We know that the distribution of [tex]X[/tex] approximates a normal distribution ⇒
[tex]X[/tex] ~ [tex]N[/tex] (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 ⇒
[tex]X[/tex] ~ [tex]N[/tex] [tex](400,50)[/tex]
The area under the normal curve between 400 and 482 represents the probability of the variable ([tex]X[/tex] in this case) to assume values between 400 and 482.
We need to calculate :
[tex]P(400<X<482)[/tex]
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
[tex]Z[/tex] ~ [tex]N(0,1)[/tex]
The probabilities of Z are in any table on internet.
To calculate [tex]P(Z\leq a)[/tex] we can use Φ(a) where Φ is the cumulative function of Z.
Solving the exercise :
[tex]P(400<X<482)[/tex] ⇒
[tex]P(\frac{400-400}{50}<Z<\frac{482-400}{50})[/tex] ⇒
[tex]P(0<Z<1.64)[/tex]
We find that [tex]P(400<X<482)=P(0<Z<1.64)[/tex]
Looking for the values of the cumulative function of Z in any table we can write :
[tex]P(0<Z<1.64)=[/tex] 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
