Normal Probability Distributions Chapter 5 Normal Probability Distributions

Chapter 5 Normal Probability Distributions Section 5-1 – Introduction to Normal Distributions and the Standard Normal Distribution The normal distribution is the most important of the continuous probability distributions. 1. Definition: A normal distribution is a continuous probability distribution for a random variable x. a. The graph of a normal distribution is called a normal curve. 2. A normal distribution has the following properties: a. The mean, median, and mode are equal (or VERY close to equal). b. The normal curve is bell-shaped and is symmetric about the mean. c. The total area under the normal curve is equal to one. d. The normal curve approaches, but never touches, the x-axis as it gets further away from the mean.

Chapter 5 Normal Probability Distributions Section 5-1 – Introduction to Normal Distributions and the Standard Normal Distribution e. The graph curves downward within one standard deviation of the mean, and it curves upward outside of one standard deviation from the mean. 1) The points where the curve changes from curving upward to curving downward are called inflection points. We know that a discrete probability can be graphed with a histogram (although we didn’t emphasize this in Chapter 4). 1. For a continuous probability distribution, you can use a probability density function (pdf). a. A probability density function has two requirements: 1) The total area under the curve has to equal one. 2) The function can never be negative.

Chapter 5 Normal Probability Distributions Section 5-1 – Introduction to Normal Distributions and the Standard Normal Distribution 2. A normal distribution can have ANY mean and ANY POSITIVE standard deviation. a. These two parameters completely determine the shape of the normal curve. 1) The mean gives the axis of symmetry. 2) The standard deviation describes how spread out (or bunched up) the data is. C. The Standard Normal Curve 1. There are infinitely many normal distributions, because there are infinitely many possible combinations of means and standard deviations. a. The standard normal distribution has a mean of zero and a standard deviation of 1.

Chapter 5 Normal Probability Distributions Section 5-1 – Introduction to Normal Distributions and the Standard Normal Distribution 1) The horizontal scale of the graph of the standard normal distribution corresponds to z-scores. a) Remember that z-scores are measures of position that indicate the number of standard deviations values lie away from the mean. 1. z = x minus the mean over the standard deviation, or 𝑧= 𝑥−𝜇 𝜎

Chapter 5 Normal Probability Distributions Section 5-1 – Introduction to Normal Distributions and the Standard Normal Distribution 2. The standard normal distribution has the following properties: a. The cumulative area under the curve is close to 0 for z-scores close to -3.49. b. The cumulative area increases as the z- scores increase. c. The cumulative area for z = 0 is 0.5000. d. The cumulative area is close to 1 for z-scores close to 3.49. 3. To find the corresponding area under the curve for any given (or calculated) z-score, there are two main methods. a. The easiest, and the one I suggest, is to use the TI-84 calculator. 1) 2nd VARS normalcdf (lower boundary, upper boundary) a) If you want the area to the left of a z-score, use -1E99 as your lower boundary and the z-score you are interested in as your upper boundary.

Chapter 5 Normal Probability Distributions Section 5-1 – Introduction to Normal Distributions and the Standard Normal Distribution b) If you want the area to the right of a z-score, use the z- score you are interested in as your lower boundary and use 1E99 as your upper boundary. c) If you want the area between two z-scores, use them both (smaller one as lower, larger one as upper).

Chapter 5 Normal Probability Distributions Section 5-1 – Introduction to Normal Distributions and the Standard Normal Distribution 4. Remember that in Section 2.4 we learned from the Empirical Rule that values lying more than 2 standard deviations from the mean are considered to be unusual. a. We also learned that values lying more than 3 standard deviations from mean are very unusual, or outliers. b. In terms of z-scores, this means that a z-score of less than -2 or greater than 2 means an unusual event. 1) A z-score of less than -3 or greater than 3 means a very unusual event. (Outlier)

III. Examples Look at Example 1 on page 241 and Example 2 on page 242 for an idea of what the graph looks like. Example 3 on Page 244 1. Find the cumulative area that corresponds to a z-score of 1.15. 2. Find the cumulative area that corresponds to a z-score of -0.24. a. 2nd VARS normalcdf Use the -1E99 already in the calculator as the low end. Use 1.15 as the high end, since this is the number you are interested in. Leave the mean and standard deviation at 0 and 1 The area to the left of 1.15 is .8749 This means that 87.49% of the curve is to the left of a z-score of 1.15.

III. Examples Look at Example 1 on page 241 and Example 2 on page 242 for an idea of what the graph looks like. Example 3 on Page 244 1. Find the cumulative area that corresponds to a z-score of 1.15. 2. Find the cumulative area that corresponds to a z-score of -0.24. b. 2nd VARS normalcdf Use the -1E99 already in the calculator as the low end. Use -0.24 as the high end, since this is the number you are interested in. Leave the mean and standard deviation at 0 and 1. The area to the left of -0.24 is .4052 This means that 40.52% of the curve is to the left of a z-score of -0.24.

Example 4 on page 246 Find the area under the standard normal curve to the left of z = -0.99 1. The percentages from the Empirical Rule, allow us to make a very good guess.   Since we want the area to the left of -0.99, and -0.99 is very close to -1.0, add the percentages to the left of -1 to get 0.158. 2nd VARS normalcdf Use the -1E99 already in the calculator as the low end. Use -0.99 as the high end, since this is the number you are interested in. Leave the mean and standard deviation at 0 and 1. The area to the left of -0.99 is 0.1611 COMPARE THIS TO OUR GUESS – our guess of .158 is very close to the answer of .161. We were within 3 tenths of a percent. Using the Empirical Rule percentages as a guide will allow us to at least decide whether our answers make sense or not!!

Example 5 on page 246 Find the area under the standard normal curve to the right of z = 1.06 2nd VARS normalcdf Use 1.06 as the low end since this is where your range begins and you are going to the right from there. Use 1E99 as the high end, since you are basically going to infinity. Leave the mean and standard deviation at 0 and 1. The area to the right of 1.06 is 0.1446 This means that 14.46% of the curve is to the right of 1.06. Again, looking at the Empirical Rule chart, adding the areas to the right of 1 standard deviation above the mean gives us .158, which is close to the actual answer.

Example 6 on page 247 Find the area under the standard normal curve between z = -1.5 and z = 1.25. 2nd VARS normalcdf Use -1.50 as the low end, for obvious reasons. Use 1.25 as the high end. Leave the mean and standard deviation at 0 and 1. The area between -1.5 and 1.25 is 0.8275. This means that 82.75% of the curve is between -1.5 and 1.25.

Your assignments are: Classwork: Pages 248–250, #9–16 All, 17-39 Odd Homework: Pages 250–252, #41–61 Odd