I'm working on some homework and I'm stuck on a question that is not making sense to me in terms of my answer.
The question states:
The following data is the last digit of the social security number of a group of students
1, 6, 9, 1, 5, 9, 0, 2, 8, 4, 0, 7, 3, 4, 2, 5, 8, 4, 2, 3, 2, 0, 0, 2, 1, 2, 7, 7, 4, 0,
0, 9, 9, 5, 3, 8, 4, 7, 4, 6, 6, 9, 0, 2, 6, 2, 9, 5, 8, 5, 1, 7, 7, 7, 8, 7, 5, 1, 8, 3,
4, 1, 9, 3, 8, 6, 6, 6, 6
a. Determine the intervals x +- s; x +- 2s; and x +- 3s.
b. Find the proportion of the measurements that lie in each of these intervals.
The standard deviation that I got was aprox 2.90647, which means my three intervals are 1.70223 to 7.51517, -1.20424 to 10.42164, and -4.11071 to 13.32811.
Which leads to my first question, is it possible to have ranges that go from negative?
Because for question b, for the 2 last intervals I get 100% for both intervals, again is that possible?
Something doesn't seem right to me.
Thank you.
The phenomenon you have described is very much possible. Indeed, for last digit of Social Security Number, it is to be expected.
A random variable $X$ with discrete uniform distribution on the integers $0$ to $9$ has mean $4.5$ and standard deviation approximately equal to $2.8723$. (This theoretical standard deviation is very close to the sample standard deviation you calculated for your data.)
Since values range from $0$ to $9$, at the theoretical level $X$ is never more than $1.57$ standard deviation units from the mean. This is roughly in line with the results you got with the sample mean and sample standard deviation from your data.
The situation is different if we draw a large sample from a normally distributed population.