Need help with statistics

Question

if the mean of 100 and a standard deviation of 15 approximately what percentage of individuals have IQ scores between 55 and 85

Answer 1

For this problem, the average µ = 100 and the standard deviation σ = 15. We want to find the corresponding z-score values for having an IQ of x = 55 and x = 85.

Since z = (x- µ) / σ, the z-score corresponding to x = 55 is as follows: z = (55-100)/15 = -3

For the upper bound, z = (85-100/15 = -1.

So, we want to find the percentage of individuals who fall in between having a z-score of -3 and -1.

Using a z-table, the probability of getting a z-score less than -1 is 0.15866.

The probability of getting a z-score less than -3 is 0.00135.

The probability encapsulated between these z-values are: 0.15866 - 0.00135 = 0.15731. Therefore, approximately 0.15731 of individuals (or 15.73% of individuals) have an IQ between 55 and 85.

Answer 2

using the empirical rule

3 standard deviations from the mean includes 99.7%

1 standard deviation: 68%

99.7-68 = 31.7

31.7/2 = 15.85% between 55 and 80

you geta slightly closer approximation using z tables or a z calculator, but the empirical rule is very close, and you can do it by just remembering 68-95-99.7 for 1, 2 and 3 standard deviations from the mean. with the empirical rule it's best to round off to nearest integer percent.