Answer:
The probability that <em>X</em> is less than 42 is 0.1271.
Step-by-step explanation:
The random variable <em>X </em>follows a Normal distribution.
The mean and standard deviation are:
E (X) = <em>μ</em> = 50.
SD (X) = <em>σ</em> = 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](https://tex.z-dn.net/?f=f_%7BX%7D%28%5Cmu%2C%20%5Csigma%29%3D%5Cfrac%7B1%7D%7B%5Csigma%5Csqrt%7B2%5Cpi%7D%7De%5E%7B-%28x-%5Cmu%29%5E%7B2%7D%2F2%5Csigma%5E%7B2%7D%7D%3B%5C%20-%5Cinfty%3CX%3C%5Cinfty%2C%5C%20%28%5Cmu%2C%20%5Csigma%29%3E0)
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}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D)
These standardized scores are known as <em>z</em>-scores and they follow normal distribution with mean 0 and standard deviation 1.
Compute the probability of (X < 42) as follows:
![P(X](https://tex.z-dn.net/?f=P%28X%3C42%29%3DP%28%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B42-50%7D%7B7%7D%29%5C%5C%3DP%28Z%3C-1.14%29%5C%5C%3D1-P%28Z%3C1.14%29%5C%5C%3D1-0.8729%5C%5C%3D0.1271)
*Use a <em>z</em>-table for the probability.
Thus, the probability that <em>X</em> is less than 42 is 0.1271.
The normal curve is shown below.