1. 1 The Normal Probability Distribution The use of tables

2. 2 The standard normal distribution Remember how we said there are many different circles and many different normal distribution? Sure you do. The z value translates any normally distributed variable into what is called the standard normal variable. Technically the picture I have on the previous screen is misleading because the z’s are a different scale than the miles, but don’t worry. In the book there is a table with z values and areas under the curve. Let’s see how to use the table. Here is one place where I want you to be extra careful when you calculate z. Round z to 2 decimal places. The z value is broken up into two parts - a.b and.0c. when added we get a.bc. For example the number 2.13 is broken up into 2.1 and.03

3. 3 Using the standard normal table The z = 2.13 means we should go down the table to 2.1 and then over to.03. The number in the table is.4834. This means the probability of getting a value of z between 0 and 2.13 is 48.34%. So, the table has the area under the curve to the left of the a positive value of interest but above the z value of 0. We may want other z’s and other areas. What do we do? 0 z This is what we get from the z table.

4. 4 The normal distribution is symmetric. Plus half the area under the curve is to the right of a z of 0 and half is to the left. So, what is the probability of a z < 0? 50%. a b What is the probability of a value of z or less? We can find the area a in the table and area b =.5, so we have tabled value +.5 0 0 z

5. 5 Say we want the area to the right of a z that is greater than 0? We know here a + b =.5 and a is in the table. So b =.5 minus the tabled value for a. a The z here would be a negative number. Say we want area a. Since the normal curve is symmetric, the area between a negative z and 0 equals the area to the left of the positive z that is the absolute value of the negative z, but greater than 0. a b 0 z z 0 a is in the table when you take the absolute value of the negative z.

6. 6 Back in the old days when I had to walk to school uphill both ways in three feet of snow, the standard normal table was all we had to calculate probabilities for a normal distribution. Now we have Microsoft Excel to make the calculations. The NORMSDIST function assumes we have a z value and we want to find the area the the left of the z - the area to the left is the cumulative probability. The function has the form =NORMSDIST(z), where z is the value we have. z can be negative in Excel. The NORMDIST function allows us to just work with the variable without getting the z and we can still have the cumulative probability. The function has the form =NORMDIST(value, mean, standard deviation, TRUE). This is an innovation of Excel over the old days.

7. 7 Sometimes we may have an area and want to know the z. The function NORMSINV asks us to give an area to the left of a value and the function will give us the z value. The form of the function is =NORMSINV(cumulative probability). The function NORMINV does the same, except not in z value form. It just give the value in the same form as the variable. The form of the function is =NORMINV(cumulative prob, mean, standard deviation)