1. Copyright © Cengage Learning. All rights reserved. Normal Curves and Sampling Distributions 7

2. Copyright © Cengage Learning. All rights reserved. Section 7.2 Standard Units and Areas Under the Standard Normal Distribution

3. 3 Focus Points Graph the standard normal distribution, and find areas under the standard normal curve.

4. 4 Standard Normal Distribution

5. 5 If the original distribution of x values is normal, then the corresponding z values have a normal distribution as well. The z distribution has a mean of 0 and a standard deviation of 1. The normal curve with these properties has a special name: Any normal distribution of x values can be converted to the standard normal distribution by converting all x values to their corresponding z values. The resulting standard distribution will always have mean  = 0 and standard deviation  = 1.

6. 6 Areas Under the Standard Normal Curve

7. 7 We have seen how to convert any normal distribution to the standard normal distribution. We can change any x value to a z value and back again. But what is the advantage of all this work? The advantage is that there are extensive tables that show the area under the standard normal curve for almost any interval along the z axis. The areas are important because each area is equal to the probability that the measurement of an item selected at random falls in this interval.

8. 8 Areas Under the Standard Normal Curve Thus, the standard normal distribution can be a tremendously helpful tool. Addition of 3 and 4 The Standard Normal Distribution (  = 0,  = 1) FIGURE 7-12

9. 9 Using a Standard Normal Distribution Table

10. 10 Using a Standard Normal Distribution Table Using a table to find areas and probabilities associated with the standard normal distribution is a fairly straightforward activity. However, it is important to first observe the range of z values for which areas are given. This range is usually depicted in a picture that accompanies the table. In this text, we will use the left-tail style table. This style table gives cumulative areas to the left of a specified z.

11. 11 Using a Standard Normal Distribution Table Determining other areas under the curve utilizes the fact that the area under the entire curve is 1. Taking advantage of the symmetry of the normal distribution is also useful. The procedures you learn for using the left-tail style normal distribution table apply directly to cumulative normal distribution areas found on calculators and in computer software packages such as Excel 2007 and Minitab.

12. 12 Example 3(a) – Standard normal distribution table Use Table 3 of Appendix to find the described areas under the standard normal curve. Find the area under the standard normal curve to the left of z = –1.00. Solution: First, shade the area to be found on the standard normal distribution curve, as shown in Figure 7-13. Figure 7-13 Area to the Left of z = –1.00

13. 13 Example 3(a) – Solution Notice that the z value we are using is negative. This means that we will look at the portion of Table 3 of Appendix for which the z values are negative. In the upper- left corner of the table we see the letter z. The column under z gives us the units value and tenths value for z. The other column headings indicate the hundredths value of z. Table entries give areas under the standard normal curve to the left of the listed z values. cont’d

14. 14 Example 3(a) – Solution To find the area to the left of z = –1.00, we use the row headed by –1.0 and then move to the column headed by the hundredths position,.00. This entry is shaded in Table 7-2. We see that the area is 0.1587. cont’d Excerpt from Table 3 of Appendix Showing Negative z Values Table 7-2

15. 15 Example 3(b) – Standard normal distribution table Find the area to the left of z = 1.18, as illustrated in Figure 7-14. Solution: In this case, we are looking for an area to the left of a positive z value, so we look in the portion of Table 3 that shows positive z values. cont’d Area to the Left of z = 1.18 Figure 7-14

16. 16 Example 3(b) – Solution Again, we first sketch the area to be found on a standard normal curve, as shown in Figure 7-14. Look in the row headed by 1.1 and move to the column headed by.08. The desired area is shaded (see Table 7-3). We see that the area to the left of 1.18 is 0.8810. cont’d Excerpt from Table 3 of Appendix Showing Positive z Values Table 7-3

17. 17 Using a Standard Normal Distribution Table Procedure:

18. 18 Using a Standard Normal Distribution Table

19. 19 Example 4(a) – Using table to find areas Use Table 3 of Appendix to find the specified areas. Find the area between z = 1.00 and z = 2.70. Solution: First, sketch a diagram showing the area (see Figure 7-16). Area from z = 1.00 to z = 2.70 Figure 7-16

20. 20 Example 4(a) – Solution Because we are finding the area between two z values, we subtract corresponding table entries. (Area between 1.00 and 2.70) = (Area left of 2.70) – (Area left of 1.00) = 0.9965 – 08413 = 0.1552 cont’d

21. 21 Example 4(b) – Using table to find areas Find the area to the right of z = 0.94. Solution: First, sketch the area to be found (see Figure 7-17). Figure 7-17 Area to the Right of z = 0.94. cont’d

22. 22 Example 4(b) – Solution (Area right of 0.94) = (Area under entire curve) – (Area left of 0.94) = 1.000 – 0.8264 = 0.1736 Alternatively, (Area right of 0.94) = (Area left of – 0.94) = 0.1736 cont’d

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