1. The Normal Distribution Lecture 20 Section 6.3.1 Fri, Oct 7, 2005

2. The Standard Normal Distribution The standard normal distribution – The normal distribution with mean 0 and standard deviation 1. The standard normal distribution – The normal distribution with mean 0 and standard deviation 1. It is denoted by the letter Z. It is denoted by the letter Z. Therefore, Z is N(0, 1). Therefore, Z is N(0, 1).

3. The Standard Normal Distribution 0123-2-3 N(0, 1) z

4. Areas Under the Standard Normal Curve What is the total area under the curve? What is the total area under the curve? What proportion of values of Z will fall below 0? What proportion of values of Z will fall below 0? What proportion of values of Z will fall above 0? What proportion of values of Z will fall above 0?

5. Areas Under the Standard Normal Curve What proportion of values of Z will fall below +1? What proportion of values of Z will fall below +1? What proportion of values of Z will fall above +1? What proportion of values of Z will fall above +1? What proportion of values of Z will fall below –1? What proportion of values of Z will fall below –1? What proportion of values of Z will fall between –1 and +1? What proportion of values of Z will fall between –1 and +1?

6. Areas Under the Standard Normal Curve It turns out that the area to the left of +1 is 0.8413. It turns out that the area to the left of +1 is 0.8413. 0123-2-3 0.8413 z

7. Areas Under the Standard Normal Curve So, what is the area to the right of +1? So, what is the area to the right of +1? 0123-2-3 0.8413 Area? z

8. Areas Under the Standard Normal Curve So, what is the area to the left of -1? So, what is the area to the left of -1? 0123-2-3 0.8413 Area? z

9. Areas Under the Standard Normal Curve So, what is the area between -1 and 1? So, what is the area between -1 and 1? 0123-2-3 0.8413 Area? 0.8413 z

10. The Three Basic Problems Find the area to the left of a: Find the area to the left of a: Look up the value for a in the table. Look up the value for a in the table. Find the area to the right of a: Find the area to the right of a: Look up the value for a; subtract it from 1. Look up the value for a; subtract it from 1. Find the area between a and b: Find the area between a and b: Look up the values for a and b; subtract the smaller value from the larger. Look up the values for a and b; subtract the smaller value from the larger. aa ba

11. Areas Under the Standard Normal Curve We will use two methods. We will use two methods. Standard normal table. Standard normal table. The TI-83 function normalcdf. The TI-83 function normalcdf.

12. The Standard Normal Table See pages 406 – 407 or pages A-4 and A-5 in Appendix A. See pages 406 – 407 or pages A-4 and A-5 in Appendix A. The entries in the table are the areas to the left of the z-value. The entries in the table are the areas to the left of the z-value. To find the area to the left of +1, locate 1.00 in the table and read the entry. To find the area to the left of +1, locate 1.00 in the table and read the entry.

13. The Standard Normal Table z.00.01.02… ::::… 0.90.81590.81860.8212… 1.00.84130.84380.8461… 1.10.86430.86650.8686… ::::…

14. The area to the left of 1.00 is 0.8413. The area to the left of 1.00 is 0.8413. That means that 84.13% of that population is below 1.00. That means that 84.13% of that population is below 1.00. 0123-2-3 0.8413

15. Standard Normal Areas What is the area to the left of 1.42? What is the area to the left of 1.42? What is the area to the right of 0.87? What is the area to the right of 0.87? What is the area between –2.14 and +1.36? What is the area between –2.14 and +1.36?

16. Let’s Do It! Let’s Do It! 6.2, p. 366 – More Standard Normal Areas. Let’s Do It! 6.2, p. 366 – More Standard Normal Areas. Use the standard normal table. Use the standard normal table.

17. TI-83 – Standard Normal Areas Press 2 nd DISTR. Press 2 nd DISTR. Select normalcdf (Item #2). Select normalcdf (Item #2). Enter the lower and upper bounds of the interval. Enter the lower and upper bounds of the interval. If the interval is infinite to the left, enter -E99 as the lower bound. If the interval is infinite to the left, enter -E99 as the lower bound. If the interval is infinite to the right, enter E99 as the upper bound. If the interval is infinite to the right, enter E99 as the upper bound. Press ENTER. Press ENTER.

18. Let’s Do It Again! Let’s Do It! 6.2, p. 366 – More Standard Normal Areas. Let’s Do It! 6.2, p. 366 – More Standard Normal Areas. Use the TI-83. Use the TI-83.

19. Other Normal Curves The standard normal table and the TI-83 function normalcdf are for the standard normal distribution. The standard normal table and the TI-83 function normalcdf are for the standard normal distribution. If we are working with a different normal distribution, say N(30, 5), then how can we find areas under the curve? If we are working with a different normal distribution, say N(30, 5), then how can we find areas under the curve?

20. Other Normal Curves For example, if X is N(30, 5), what is the area to the left of 35? For example, if X is N(30, 5), what is the area to the left of 35? 30354045252015

21. Other Normal Curves For example, if X is N(30, 5), what is the area to the left of 35? For example, if X is N(30, 5), what is the area to the left of 35? 30354045252015

22. Other Normal Curves For example, if X is N(30, 5), what is the area to the left of 35? For example, if X is N(30, 5), what is the area to the left of 35? 30354045252015 ?

23. Other Normal Curves To determine the area, we need to find out how many standard deviations 35 is above average. To determine the area, we need to find out how many standard deviations 35 is above average. Since  = 30 and  = 5, we find that 35 is 1 standard deviation above average. Since  = 30 and  = 5, we find that 35 is 1 standard deviation above average. Thus, we may look up 1.00 in the standard normal table and get the correct area. Thus, we may look up 1.00 in the standard normal table and get the correct area.

24. Other Normal Curves For example, what is the area to the left of 35? For example, what is the area to the left of 35? 30354045252015 0.8413 0123-2-3 X Z

25. Z-Scores Z-score, or standard score, of an observation – The number of standard deviations from the mean to the observed value. Z-score, or standard score, of an observation – The number of standard deviations from the mean to the observed value. Compute the z-score of x as Compute the z-score of x asor Equivalently Equivalentlyor

26. Areas Under Other Normal Curves If a variable X has a normal distribution, then the z-scores of X have a standard normal distribution. If a variable X has a normal distribution, then the z-scores of X have a standard normal distribution. If X is N( ,  ), then (X –  )/  is N(0, 1). If X is N( ,  ), then (X –  )/  is N(0, 1). That is, That is,

27. Example Let X be N(30, 5). Let X be N(30, 5). What proportion of values of X are below 38? What proportion of values of X are below 38? Compute z = (38 – 30)/5 = 8/5 = 1.6. Compute z = (38 – 30)/5 = 8/5 = 1.6. Find the area to the left of 1.6 under the standard normal curve. Find the area to the left of 1.6 under the standard normal curve. Answer: 0.9452. Answer: 0.9452. Therefore, 94.52% of the values of X are below 38. Therefore, 94.52% of the values of X are below 38.

28. Let’s Do It! Let’s Do It! 6.3, p. 367 – IQ Scores. Let’s Do It! 6.3, p. 367 – IQ Scores.

29. The “68-95-99.7 Rule” The 68-95-99.7 Rule: For any normal distribution N( ,  ), The 68-95-99.7 Rule: For any normal distribution N( ,  ), 68% of the values lie within  of . 68% of the values lie within  of . 95% of the values lie within 2  of . 95% of the values lie within 2  of . 99.7% of the values lie within 3  of . 99.7% of the values lie within 3  of .

30. The “68-95-99.7 Rule” Equivalently, Equivalently, 68% of the values lie in the interval 68% of the values lie in the interval [  – ,  +  ], or   . 95% of the values lie in the interval 95% of the values lie in the interval [  – 2 ,  + 2  ], or   2 . 99.7% of the values lie in the interval 99.7% of the values lie in the interval [  – 3 ,  + 3  ], or   3 . More precisely, the percentages are 68.27%, 95.45%, and 99.73%. More precisely, the percentages are 68.27%, 95.45%, and 99.73%.

31. The Empirical Rule The well-known Empirical Rule is similar, but more general. The well-known Empirical Rule is similar, but more general. If X has a “mound-shaped” distribution, then If X has a “mound-shaped” distribution, then Approximately 68% lie within  of . Approximately 68% lie within  of . Approximately 95% lie within 2  of . Approximately 95% lie within 2  of . Nearly all lie within 3  of . Nearly all lie within 3  of .

32. Let’s Do It! Let’s Do It! 6.5, p. 369 – Pine Needles. Let’s Do It! 6.5, p. 369 – Pine Needles. Let’s Do It! 6.6, p. 369 – Last Longer? Let’s Do It! 6.6, p. 369 – Last Longer?