The Standard Normal Distribution Lecture 25 Section 6.3.1 Mon, Mar 1, 2004

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

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

Areas Under the Standard Normal Curve What proportion of values of Z will fall below 0? 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 below –1?

Areas Under the Standard Normal Curve We need to find the area under the curve to the left of +1. -3 -2 -1 1 2 3

Areas Under the Standard Normal Curve This is too hard to calculate by hand. We will use Standard normal table. The TI-83 function normalcdf.

The Standard Normal Table See pages 372 – 373 or pages 942 – 943. 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.

The Standard Normal Table z .00 .01 .02 … : 0.9 0.8159 0.8186 0.8212 1.0 0.8413 0.8438 0.8461 1.1 0.8643 0.8665 0.8686

The Standard Normal Table The area to the left of +1 is 0.8413. 0.8413 -3 -2 -1 1 2 3

Standard Normal Areas What is the area to the right of +1? What is the area to the left of -1? What is the area to the right of -1? What is the area between -1 and +1?

Let’s Do It! Let’s do it! 6.1, p. 332 – More Standard Normal Areas. Use the standard normal table.

The TI-83 and Standard Normal Areas Press 2nd DISTR. Select normalcdf (Item #2). Enter the lower and upper bounds of the interval. If the interval is infinite to the left, enter -99 as the lower bound. If the interval is infinite to the right, enter 99 as the upper bound.

Let’s Do It Again! Let’s do it! 6.1, p. 332 – More Standard Normal Areas. Use the TI-83.

Standard Normal Areas on the TI-83 Press ENTER. Examples: normalcdf(-99, 1) = 0.8413447404. normalcdf(1, 99) = 0.1586552596. normalcdf(-99, -1) = 0.1586552596. normalcdf(-1, 99) = 0.8413447404. normalcdf(-1, 1) = 0.6826894809.

Assignment Page 341: 1, 2, 7a, 7b, 7c, 7d, 7e, 7f, 7g.

Areas Under Other Normal Curves Let X be N(, ). Then (X – )/ is N(0, 1). That is, Z = (X – )/. The value of Z is called the z-score or standard score of X.

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

Let’s Do It! Let’s do it! 6.2, p. 333 – IQ Scores.

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

The “68-95-99.7 Rule” Equivalently, 68% of the values lie in the interval [ – ,  + ], or   . 95% of the values lie in the interval [ – 2,  + 2], or   2. 99.7% of the values lie in the interval [ – 3,  + 3], or   3.

The Empirical Rule The well-known Empirical Rule is similar, but more general. If X has a “mound-shaped” distribution, then Approximately 68% lie within  of . Approximately 95% lie within 2 of . Approximately 99.7% lie within 3 of .

Let’s Do It! Let’s do it! 6.5, p. 335 – Last Longer?

Assignment Page 341: Exercises 3, 4, 5, 9, 11, 14, 24, 25, 29, 31.