Day 12

Math 216: Statistical Thinking

Bastola

Normal Distribution

\[ f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

Standard Normal Distribution

\[ f(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} \]

Example 1: Find the probability that the standard normal random variable \(z\) is less than 1.28 ; that is, find \(P(z<1.28)\)

Example 2: Find the probability that the standard normal random variable \(z\) is exceeds 1.64 ; that is, find \(P(z>1.64)\)

Example 3: Find the probability that the standard normal random variable \(z\) less than -1.64 ; that is, find \(P(z<-1.64)\)

Example 4: Find the probability that the standard normal random variable \(z\) exceeds -1.28 ; that is, find \(P(z>-1.28)\)

Example 5: Find the probability that the standard normal random variable \(z\) lies between -1.64 and 1.28 ; that is, find \(P(-1.64<z<1.28)\)

Example 6: Find the probability that the standard normal random variable \(z\) exceeds 1.96 in absolute value; that is, find \(P(|z|>1.96)\)

Converting Normal Variable to Standard Normal

To utilize the standard normal distribution effectively in statistical calculations, we convert a normal random variable \(x\) with any mean \(\mu\) and standard deviation \(\sigma\) to a standard normal variable \(z\).

Z-score

\[ Z = \frac{x - \mu}{\sigma} \]

Example:

To find the probability of \(x\) being less than a particular value \(x_0\), use: \[ P(x \leq x_0) = P\left(Z \leq \frac{x_0 - \mu}{\sigma}\right) \]

Example 7: Assume that the length of time, x , between charges of a cellular phone is normally distributed with a mean of 10 hours and a standard deviation of 1.5 hours. Find the probability that the cell phone will last between 8 and 12 hours between charges.

Example 8: Suppose the scores \(x\) on a college entrance examination are normally distributed with a mean of 550 and a standard deviation of 100 . A certain prestigious university will consider for admission only those applicants whose scores exceed the 90th percentile of the distribution. Find the minimum score an applicant must achieve in order to receive consideration for admission to the university.

Example 9: Find \(z_0\) such that \(P\left(-z_0 \leq z \leq z_0\right)=0.95\)