Question

(12.5) Describe the procedure to determine the percent of data between any two values in a normal distribution.

Answer 1

Finding the percentage: In order to compute percentages under a normal distribution, you need to standartize every given value

=>For example, to find P(x < b) under the normal distribution N(µ, σ), you first standartize b to b−µ /σ .

=>Then you need to find P(z < b−µ /σ ). Look up the value of b−µ/ σ in table A (“Standard normal probabilities”).

=>The corresponding number in the table is the required proportion. To convert to percentages, multiply by 10

=> To find other proportions, we use geometric facts that P(a < z < b) = P(z < b) − P(z < a) (see the picture) and P(z > a) = 1 − P(z < a).

=>Example, continued. Consider the normal distribution N(100, 10).

=>To find P(97.1 < x < 105.3), standartize first: P(97.1 < x < 105.3) = P 97.1 − 100 10 < z < 105.3 − 100 10 = P(−0.29 < z < 0.53)

=> Then P(−0.29 < z < 0.53) = P(z < 0.53) − P(z < −0.29).

=>The last two proportions can be found in Table A: P(z < 0.53) = .7019 and P(z < −0.29) = .3859 (row −0.2, column 0.09).

=>Thus P(97.1 < x < 105.3) = .7019 − .3859 = .3160 or 31.6%.

From percentages to values:

=>There is another kind of problems: given a percentage, find the corresponding boundary value.

=>For example, given the percentage P(x < b) = P, what is b? Here to find b, we look up P or the value closest to P in the table and find the corresponding z-score.

=>Then, we need to solve z = b−µ σ for b. Algebra shows that b = zσ + µ. Example, continued. Consider the normal distribution N(100, 10).

=>What values lie in the lower 80% of the data? We need to find b such P(x < b) = 80%. First we find the z-score Z such that P(z < Z) = 80%.

=>The table does not contain 0.8; the closest number is 0.7995. It lies in the row 0.8 and column 0.04. Thus the z-score of b is approximately 0.84: 0.84 = b − 100/ 10 .

Hence b − 100 = 0.84 × 10 = 8.4 and b = 100 + 8.4 = 108.4. We conclude that the lower 80% of this distribution is formed by values below 108.4