THE STANDARD NORMAL DISTRIBUTION SECTION 6.2 THE STANDARD NORMAL DISTRIBUTION

NORMAL PROBABILITY DISTRIBUTION The mean, median and mode are the same. The distribution is bell-shaped and symmetrical around the mean. The total area under the curve is equal to 1. The left side and the right side of the normal probability distribution extend indefinitely, never quite touching the horizontal axis.

NORMAL DISTRIBUTION If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped and it can be described by the equation we say that is has a normal distribution.

STANDARD NORMAL DISTRIBUTION Is a normal distribution with the parameters of µ = 0 and σ = 1. The total area under its density curve is equal to 1.

FINDING PROBABILITIES This section will have you find the probability / area under the curve to the z – score. There are 5 unique style questions P(a < z < b) P(a < z ) P(z < b) Zα P%

P(a < z < b) The probability that the z score is between a and b.

P(a < z) The probability that the z score is greater than a.

P(z < b) The probability that the z score is less than b. b

Zα The z-score with an area of α to its right.

P% The z-score with an area from 0 to the given %. Change % to decimal, % = α .

Calculator – Inequality How to find the probability with a TI Go to folder DISTR (2nd Vars) Press 2 (normalcdf) – gives percentage of area under a standard normal distribution. Normalcdf(lower bound, upper bound) OR Normalcdf( lower bound, upper bound, mean , standard deviation)

TABLE - Inequality Given the z score table the graph at the top of the page shows you the probability / area. Understand different z scores shade different sides, so be careful. The probabilities / areas are on the inside. The z – score is broken up into two parts The whole and the tenth number are on the side The hundredth number is on the top Match the two together to find the prob / area

EXAMPLE Find the probability of a normal standard deviation between -0.62 and 1.78

EXAMPLE Find the probability of a normal standard deviation greater than -1.04

EXAMPLE Find the probability of a normal standard deviation less than -1.04

Calculator - Zα Subtract : 1 – α Distr (2nd vars) InvNorm( Type in 1 – α enter

Table - Zα Look at the shaded region of the z table This α connects with a shaded region to the right, if you need the shaded region to the left then subtract from 1, 1 – α Find the value inside the z score table Look up and to the left and find the z value

EXAMPLE Evaluate z0.025

Calculator - P% Distr button (2nd vars) Press number 3 invNorm InvNorm (area → decimal no %)

Table - P% Look at the middle section of the z score table Convert & to decimal Find the probability If exact probability is not there adjust accordingly Move up and left for the two part and combine them.

EXAMPLE Evaluate P90

EXAMPLE Evaluate z0.025