STA 2023 Sections 5.1 and 5.2 Finding Probabilities for Normal Distributions.
Properties of a Normal Distribution • A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.
Properties: • The mean, median, and mode are equal. • The normal curve is bell-shaped and is symmetric about the mean. • The total area under the normal curve is equal to 1. • The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean. • The x-axis is a horizontal asymptote to the curve • The graph contains points of inflection located 1 standard deviation away from the mean. • These are points where the graph changes the way it curves.
Finding Areas Under the Standard Normal Curve • To find areas under the Standard Normal Curve, we will be using the calculator. • Press 2nd, Vars, then 2:normalcdf( • Syntax: normalcdf(lower bound, upper bound) • The lower bound and the upper bound correspond to the area of the standard normal curve that we are finding. • Always draw the standard normal curve and shade in the area you are looking for so that you clearly find your lower and upper bound. • If a tail is use as a bound for the area, use -10000 for the lower bound or 10000 for the upper bound.
Example 1: Find the area under the standard normal curve to the left of z = 2.13. • Answer: normalcdf(-10000,2.13) = .9834 • Example 2: Find the area under the standard normal curve to the right of z = -1.16. • Answer: normalcdf(-1.16,10000) = .8770 • Example 3: Find the area under the standard normal curve between z = -2.17 and z = -1.35. • Answer: normalcdf(-2.17, -1.35) = .0735 • Example 4: Find the area under the standard normal curve to the left of z = -0.82 or to the right of z = 1.17. • Answer: normalcdf(-10000,-0.82)+normalcdf(1.17,10000) = .3271
Probability and Normal Distributions • We can find the probability of any normal distribution by converting the data into the standard normal distribution using the z-score formula. • The area under the standard normal curve is equal to the probability of an event happening in the normal distribution.
Example 6: The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $12. A utility bill is randomly selected. • Find the probability that the utility bill is less than $70. • Answer: .0062 • Find the probability that the utility bill is between $90 and $120. • Answer: .7493 • If we look at a group of 150 utility bills, how many of those bills will be between $90 and $120? • Answer: 112 utility bills. • Find the probability that the utility bill is more than $140. • Answer: .0004