Normal Random Variables (3 of 6)

 

Learning Objectives

  • Use a normal probability distribution to estimate probabilities and identify unusual events.

Example

The Empirical Rule in a Context

Suppose that foot length of a randomly chosen adult male is a normal random variable with mean [latex]\mathrm{μ}=11[/latex] and standard deviation [latex]\mathrm{σ}=1.5[/latex] . Then the empirical rule lets us sketch the probability distribution of X as follows:

Probability distribution of X (foot length), where the curve is black, and the data is in blue

  • (a) What is the probability that a randomly chosen adult male will have a foot length between 8 and 14 inches?
  • Answer: 0.95, or 95%
  • (b) An adult male is almost guaranteed (0.997 probability) to have a foot length between what two values?
  • Answer: 6.5 and 15.5 inches
  • (c) The probability is only 2.5% that an adult male will have a foot length greater than how many inches?
  • Answer: 14 inches

Ninety-five percent of the area is within 2 standard deviations of the mean, so 2.5% of the area is in the tail above 2 standard deviations. The x-value 2 standard deviations above the mean is 14 inches.

Normal curve showing 95% of area is within 2 SD of the mean

Now you should try a few: questions (d), (e), and (f) are presented in the Learn By Doing activity. Use the figure preceding question (a) to help you.

Learn By Doing

Comment

Notice that there are two types of problems we may want to solve: those like (a) and, from the Learn By Doing activity, (d) and (e), in which a particular interval of values of a normal random variable is given and we are asked to find a probability; and those like (b), (c), and, from the Learn By Doing, (f), in which a probability is given and we are asked to identify values of the normal random variable.