Statistic for Business, Lecture notes of Statistics

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STATISTICS FOR BUSINESS

II

Doç. Dr. Yüksel Akay ÜNVAN

THE NORMAL DISTRIBUTION

The place of histograms will be taken by continuous curves, as in Figure 9.1, picturing them mentally as being approximated by histograms with narrower and narrower classes.

Continuous curves such as the one shown on the right in Figure 9.2 are the graphs of functions called probability densities, or informally, continuous distributions

The graph of a normal distribution is a bell-shaped curve that extends indefinitely in both directions. Although this may not be apparent from a small drawing like that of figure 9.

In all our work with normal distributions, we shall be concerned only with areas under their curves—so-called normal-curve areas—and such areas are found in practice from tables such as Table I at the end of the book. As it is physically impossible, but also unnecessary, to construct separate tables of normal-curve areas for all conceivable pairs of values of μ and σ we tabulate these areas only for the normal distribution with μ = 0 and σ = 1, called the standard normal distribution. Then, we obtain areas under any normal curve by performing the change of scale (see Figure 9.7) that converts the units of measurement from the original scale, or x-scale, into standard units, standard scores, or z-scores, by means of the formula 8

Example 9.4 Find the standard-normal-curve area a) to the left of z = 0.94; b) to the right of z = -0.65; c) to the right of z = 1.76; d) to the left of z = -0.85; e) between z = 0.87 and z = 1.28; f) between z = -0.34 and z = 0.62.

EXAMPLE 9.5. If a random variable has the normal distribution with μ= 10 and σ= 5, what is the probability that it will take on a value on the interval from 12 to 15? (Table 1 refers to Z-table)