Standard Normal Distribution - Probability and Statistics - Lecture Slides, Slides of Probability and Statistics

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Chapter 6

The Normal Distribution

In this handout:

• The standard normal distribution

• Probability calculations with normal distributions

• The normal approximation to the binomial

Figure 6.8 (p. 231) The standard normal curve. Figure 6.9 (p. 232) Equal normal tail probabilities.

It is customary to denote the standard normal variable by Z.

Figure 6.11 (p. 233) Normal probability of an interval.

P[-.155 < Z < 1.6] = P[Z < 1.6] - P[Z < -.155] = .9452 - .4384 = .5068 Docsity.com

Figure 6.12 (p. 233) Normal probabilities for Example 3.

P[Z < -1.9 or Z > 2.1] = P[Z < -1.9] + P[Z > 2.1] = .0287 + .0179 = .0466 Docsity.com

Determining z for given equal tail areas

• Problem : Obtain the value of z for which P[-z < Z < z] = 0.
• Solution : From the symmetry of the curve, P[ Z < -z] = P[ Z > z] =. From the table, P[ Z < -1.645] = 0.05. Thus z = 1.

Converting a normal probability to a standard

normal probability

Figure 6.16 (p. 241) The binomial distributions for p = .4 and n = 5, 12, 25.

When the success probability p of is not too near 0 or 1 and the number of trials is large, the normal distribution serves as a good approximation to the binomial probabilities.

How to approximate the binomial probability by a normal? The normal probability assigned to a single value x is zero. However, the probability assigned to the interval x-0.5 to x+0.5 is the appropriate comparison (see figure). The addition and subtraction of 0.5 is called the continuity correction.