Question

EXAMPLE 5 Intelligence Quotient (10) scores are distributed normally with mean 100 and standard deviation 15. The corresponding density function is shown in the figure 0.02+ (a) What percentage of the population has an IQ score between 85 and 115? 001+ A A 60 80 100 120 140 (b) What percentage of the population has an IQ above 1607 SOLUTION (a) Since IQ scores are normally distributed, we use the probability density function with y = 100 and 15: Video Example Tutorial Online Textbook p(855 X 5115) - 1672 2017 Recall that the function ye doesn't have an elementary antiderivative, so we can't evaluate the integral exactly. But we can use the numerical integration capability of a calculator or computer or the Midpoint Rule or Simpson's Rule) to estimate the integral. Doing so, we find that P(85 SX S115) So about % of the population has an IQ between 85 and 115; that is within one standard deviation of the mean. (b) The probability that the IQ score of a person chosen at random is more than 160 is 1 P(X>c) - 1 152 To avoid the improper integral we could approximate it by the integral from to 200. (It's quite safe to say that people with an IQ over 200 are extremely rare.) Then PX>c) -

Answer 1

6 Ans!- (a) P185 $*$115) 10.68268 so about 68.27%, of the popelation has a lo between 85 and 115, b) To arold the comproper integral we could approxieincete it by the to 200 integral foon (160 .. Pix > 160) 10.00003 031677 Therefore about 0.0031% of the popielcetion has aos in over 160 CS Scanned with CamScanner