Chapter 5 Normal Probability Distributions Section 5-2 – Normal Distributions: Finding Probabilities  A. On the STANDARD normal curve, the mean is always 0 and the standard deviation is always 1, and we always use z-scores. 1. There are an infinite number of possible normal curves, each with its own mean and standard deviation. 2. To find the probability of any particular x value in one of these other normal distributions, we will use the same distribution on the calculator, simply changing the mean and standard deviation to match the data.

Example 1 on page 253 A survey indicates that people use their computers an average of 2.4 years before upgrading to a new machine. The standard deviation is 0.5 year. A computer owner is selected at random. Find the probability that he or she will use it for fewer than 2 years before upgrading. Assume that the variable x is normally distributed. 0.0015 0.0215 0.136 0.341 0.341 0.136 0.0215 0.0015 0.9 1.4 1.9 2.4 Since 2 is slightly more than 1.9, we expect our answer to be between .1575 and .5, but closer to .1575.

Example 1 on page 253 A survey indicates that people use their computers an average of 2.4 years before upgrading to a new machine. The standard deviation is 0.5 year. A computer owner is selected at random. Find the probability that he or she will use it for fewer than 2 years before upgrading. Assume that the variable x is normally distributed. 2nd VARS normalcdf Use -1E99 as the low end. Use 2 as the high end, since this is the number you are interested in. Change the mean to 2.4 Change the standard deviation to 0.5 The area to the left of 2 is 0.2119 This matches our guess of between 0.1575 and .5 and closer to .1575, so we can be confident in this answer.

Example 3 on page 255 Assume that the cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level (a) is less than 175; (b) is between 175 and 225; (c) is more than 225? 0.0015 0.0215 0.136 0.341 0.341 0.136 0.0215 0.0015 140 165 190 215 240 265 290 Since 175 is between 165 and 190, we can expect our probability to be between .023 and .1575. Since 175 is closer to 165 than it is to 190, we can also expect our answer to be closer to .023 than .1575.

Example 3 on page 255 Assume that the cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level (a) is less than 175; (b) is between 175 and 225; (c) is more than 225? (a) 2nd VARS normalcdf Use -1E99 as the low end. Use 175 as the high end Change the mean to 215 Change the standard deviation to 25 The area to the left of 175 is .0548 This fits our expectations of between .023 and .1575 and closer to .023, so we can be comfortable with it.

Example 3 on page 255 Assume that the cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level (a) is less than 175; (b) is between 175 and 225; (c) is more than 225? 0.0015 0.0215 0.136 0.341 0.341 0.136 0.0215 0.0015 140 165 190 215 240 265 290 To go from 175 to 225, we cross the mean. We also cross all of the 34.1% between 190 and 215, as well as much of that between 215 and 240. It would be safe to expect an answer of at least .5, and probably more than that.

Example 3 on page 255 Assume that the cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level (a) is less than 175; (b) is between 175 and 225; (c) is more than 225? (b) 2nd VARS normalcdf Use 175 as the low end. Use 225 as the high end. Change the mean to 215 Change the standard deviation to 25 The area between 175 and 225 is 0.6006. Again, our guess was a rough one, but .6006 is not totally unexpected and makes sense.

Example 3 on page 255 Assume that the cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level (a) is less than 175; (b) is between 175 and 225; (c) is more than 225? 0.0015 0.0215 0.136 0.341 0.341 0.136 0.0215 0.0015 140 165 190 215 240 265 290 To go from 225 and up, we don’t cross the mean, so we know the answer will be less than .5. It will also be greater than .1575, since it is less than 240.

Example 3 on page 255 Assume that the cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level (a) is less than 175; (b) is between 175 and 225; (c) is more than 225? (c) 2nd VARS normalcdf Use 225 as the low end. Use 1E99 as the high end. Change the mean to 215 Change the standard deviation to 25 The area to the right of 225 is .3446. This fits with our expectation of between .158 and .5, so we’re happy.

Your assignments are: Classwork: Pages 256–258, #1–12 All, 14–20 Evens Homework: Pages 258–259, #21–30 All