8. The Standard Normal Distribution

l. determine the probability that a normally distributed random variable lies inside a given interval;

## m. define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution;

What is the standard normal distribution? The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is denotes as N(0, 1).

What are the standard deviations and confidence intervals for a standard normal distribution? This is related to the empirical rule. Mean = 0 First standard deviation = 1 with 68% confidence Second standard deviation = 1.96 with 95% confidence Third standard deviation = 2.58 with 99.7% confidence

What is the formula to transform a normal distribution into a standard normal distribution? \(z = \frac{X - \mu}{\sigma}\) - X = score from the original normal distribution - \mu = mean of the original normal distribution - \sigma = standard deviation of the original normal distribution

What is a Z-score? How do you calculate it? - The Z-score is the number of standard deviations away from the mean for a normal distribution. If the mean is 5, the standard deviation is 10, then the z-score for x=20 would be 1.5. - You could use the formula (X - mean)/std-dev

What is a Z-table and how do you use it? - A z-table is a table of values that indicate the percentage of area under a normal distribution that lies below the given z-score (# of std dev from the mean). - The z-table has integers down the left and decimals across the top. Just match your given score down and across to find the percent of area that lies below.