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

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Presentation on theme: "STA 2023 Sections 5.1 and 5.2 Finding Probabilities for Normal Distributions."— Presentation transcript:

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

2 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.

3  Properties: 1. The mean, median, and mode are equal. 2. The normal curve is bell-shaped and is symmetric about the mean. 3. The total area under the normal curve is equal to The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean. 1. The x-axis is a horizontal asymptote to the curve 5. 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.

4 The Standard Normal Distribution 

5 Finding Areas Under the Standard Normal Curve  To find areas under the Standard Normal Curve, we will be using the calculator.  Press 2 nd, 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.

6  Example 1: Find the area under the standard normal curve to the left of z =  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 =  Answer: normalcdf(-10000,-0.82)+normalcdf(1.17,10000) =.3271

7 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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9  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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