Question

Draw 1000 Uniform random variable. You know the true mean. You can use the standard deviation of your data or you can calculate the true standard deviation and use that. How often does your data fall within 2 standard deviations of the true mean? How about 1 standard deviation?

Repeat the above with 1000 Normal Random variables

Draw 9 Normal Random variables and calculate the average. We will work with the averages.Repeat 1000 times so we have 1000 averages of 9 draws. What range does 95% of the data fall into? What about 68%? According to our math this range should be 1/3 the size of what you saw in (b). Repeat with 16 draws per group. Does this cut the range to 1/4 of the old range?

R would be ideal, but Excel or SPSS are also fine

Answer 1

The true mean of the uniform random variable in the range (0,1) is 0.5 and the standard deviation is 0.288. 59.1% of observation fall within one standard deviation of the true mean and 100% observations fall between 2 standard deviations in the case of uniform distribution because the interval observed for 2 standard deviations is (-0.077,1.077). A screen shot of the analysis is provided below

The same experiment is repeated for normal distribution by simulating 1000 observations from N(0,1). and the screen shot of the analysis is given below. The mean calculated from the data is -0.073 and standard deviation is 0.9644. It is found that 68% of the observation fall within one standard deviation and 96.7% of observations within two standard deviations.

after drawing 9 random variables 1000 times the interval (-0.3364,0.3392) is the interval with one standard deviation Where as earlier when there is no sampling the interval was (-1,1). So when we take average from 9 random variables and repeated this 1000 times the interval reduces to 1/3 of the previous interval. a screen shot of the analysis is provided below