Statistics 3011 (Geyer and Jones, Spring 2006) Examples: Normal Distributions

Contents

Standard Normal Distribution

Direct Look-Up

For our example we redo Example 3.5 in the Textbook (Moore)

Comments

Inverse (Backward) Look-Up

Inverse look-up, which the book calls backward look-up (p. 72) or finding a value given a proportion (p. 71) is like playing Jeopardy (given the answer find the question). If we write

p = Pr(Z < b)

then the preceding section is about given b find p (and related questions) and this section is about given p find b (and related questions).

The following box answers the following two questions

  1. What is the point z such that the area under the standard normal curve to the left of z is 0.95?
  2. What is the point z such that the area under the standard normal curve to the right of z is 0.25?

Comments

General Normal Distribution

The methodology for looking up in tables involves standardization and unstandardization, because we have only a table of the standard normal distribution. With R, those steps are unnecessary. It is as if R has tables for all normal distributions. Don't bother with the standardization and unstandardization.

Direct Look-Up

This is Example 3.6 in the book. The question stripped of everything irrelevant to the calculation is: for a (general) normal distribution with mean 170 and SD 30 what is the probability to the right of 240?

The book is wrong. It says the answer is 0.0099 but R says the answer is 0.0098. Close enough for homework, I suppose.

This is Example 3.7 in the book. The question stripped of everything irrelevant to the calculation is: for a (general) normal distribution with mean 170 and SD 30 what is the probability between 170 and 240 (for the interval 170 < x < 240)?

The book is wrong again. It says the answer is 0.4901 but R says the answer is 0.4902. Close enough for homework, I suppose.

Inverse Look-Up

This is Example 3.8 in the book. The question stripped of everything irrelevant to the calculation is: for a (general) normal distribution with mean 504 and SD 111 what is the x such that the probability above x is 0.10? (The book says 10% but we always convert percents to pure number to deal with them as probabilities).

The book is wrong again. It says the answer is 646.1 but R says the answer is 646.25. Close enough for homework, I suppose.

All of these inaccuracies in the book's calculations are partly due to the limitations of the tables (not enough significant figures there) and partly do to too much rounding in intermediate calculations. R just does it right with no fuss.