If one score is randomly selected from a normal distribution with µ = 100 and σ = 20, the probability of obtaining a score less than X = 70 is p = 0.0013.

Answer 1

93.32% of the scores will fall within 1.5 standard deviations from the mean.

70 is 1.5 standard deviations from the mean. then look up 1.5 in z tables or on a calculator to get 93.32%

Then subtract 93.32 from 100 to get 6.68% Divide that by 2 to get 3.34% convert to decimal form: 0.0334 is the probability of getting less than 70 given mean 100 and standard deviation 20 with a normal distribution.

Just from the 68-95-99 empirical rule, you know 95% falls within 2 standard deviations, and 68% fall within 1 standard deviation. You know 1.5 standard deviations is somewhere between those two percentages. Smallest possible probability for less than 70 is 1/2 x 95% or 2.5% = 0.025 = p 0.0334 is a little more, as 1.5 is less than 2 standard deviations. 95% falls within 2 standard deviations. The remaining 5% is split with half over 1.5 above the mean, and the other half, the other 2.5% below 1.5 standard deviations from the mean.