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**Sections 5.1 and 5.2 Finding Probabilities for Normal Distributions.**

STA 2023 Sections 5.1 and 5.2 Finding Probabilities for Normal Distributions.

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**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.

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**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.

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**The Standard Normal Distribution**

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**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 for the lower bound or for the upper bound.

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**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 = Answer: normalcdf(-1.16,10000) = .8770 Example 3: Find the area under the standard normal curve between z = and z = Answer: normalcdf(-2.17, -1.35) = .0735 Example 4: Find the area under the standard normal curve to the left of z = or to the right of z = 1.17. Answer: normalcdf(-10000,-0.82)+normalcdf(1.17,10000) = .3271

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**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.

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**Find the probability that the utility bill is less than $70. **

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

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