What is the 68-95-99.7 rule and could you demonstrate on these problems the mean of 70 and a standard deviation of 15 less than 55 less than 40 less than 85 less than 100 greater than 85?

Answer 1

The rule applies to the normal distribution.

For any normal distribution, 68% of the observations lie within 1 standard deviation (SD) either side of the mean that is, between (mean - SD) and (mean + SD); 95% of the observations lie between (mean - 2*SD) and (mean + 2*SD); 99.7% between (mean - 3*SD) and (mean + 3*SD).

To find the probability that a normally distributed random variable, with mean = m and SD = s, you calculate its z-score = (X - m)/s and look up the relevant probability value in tables.

In doing this, you may have to use the symmetry of the normal distribution or the probability of complementary events (or both). This will depend on what exactly is tabulated.

Mean 70, SD = 15

Pr[X < 55] = Pr[Z < (55 - 70)/15] = Pr[Z < -1] = 0.5*(1-0.68) = 0.5*0.32 = 0.16

Pr[X < 40] = Pr[Z < (40 - 70)/15] = Pr[Z < -2] = 0.5*(1-0.95) = 0.5*0.05 = 0.025

Pr[X < 85] = Pr[Z < (85 - 70)/15] = Pr[Z < 1] = 1 - 0.16 = 0.84

Pr[X < 100] = Pr[Z < (100 - 70)/15] = Pr[Z < 2] = 1 - 0.025 = 0.975

Pr[X > 85] = 1 - Pr[X < 85] = 1 - 0.84 = 0.16