Question

Assume the random variable X is normally distributed with mean mu equals 50 and standard deviation sigma equals 7. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded.

Answer 1

Answer:

The probability that X is less than 42 is 0.1271.

Step-by-step explanation:

The random variable X follows a Normal distribution.

The mean and standard deviation are:

E (X) = μ = 50.

SD (X) = σ = 7.

A normal distribution is continuous probability distribution.

The Normal probability distribution with mean µ and standard deviation σ is given by,

$f_{X}(\mu, \sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^{2}/2\sigma^{2}};\ -\infty<X<\infty,\ (\mu, \sigma)>0$

To compute the probability of a Normal random variable we first standardize the raw score.

The raw scores are standardized using the formula:

$z=\frac{x-\mu}{\sigma}$

These standardized scores are known as z-scores and they follow normal distribution with mean 0 and standard deviation 1.

Compute the probability of (X < 42) as follows:

$P(X<42)=P(\frac{x-\mu}{\sigma}<\frac{42-50}{7})\\=P(Z<-1.14)\\=1-P(Z<1.14)\\=1-0.8729\\=0.1271$

*Use a z-table for the probability.

Thus, the probability that X is less than 42 is 0.1271.

The normal curve is shown below.