From a sample of size n = 100, the following descriptive measures were calculated: median=23, mean=20, standard deviation=5, range=35; 75 sample values are between 10 and 30; and 99 sample values are between 5 and 35. If you knew the sample mean, median, and standard deviation were correct, which of the following conclusions might you draw?
(a) The distribution is skewed to the right because the median exceeds the mean.
(b) The range must have been calculated incorrectly because it should not be seventimes the standard deviation’s value.
(c) The number of sample values between 10 and 30 was miscounted.
(d) The number of sample values between 5 and 35 must have been miscounted because all 100 values must be in this interval.
a) it is not this one because if media exceeds mean, then it should be skewed left.
b) range should not affect standard deviation, I think, therefore this one is wrong
b) By the tchebyshef's theorem, 75% should of measurements should be within 2 standard deviations, which turns out to be exactly correct. $20 \pm 2(5) = (10,30)$ which is 75% of our measurements
d) this is where I need clarification. I believe this is the true conclusion because by the empirical rule, 99.7% (which constitutes as 100%) of the sample should cover 3 standard deviations. $20 \pm 3(5) = (5,35)$ so 3 standard deviations should cover 100% of our measurements correct? Therefore this should be 100 and not 99.
Any advices.
For d you are making two errors. First, $99.7\%$ is not $100\%$. Events outside $3\sigma$ do happen, though rarely. Second, as you are not given that the distribution is normal, the best you can do is Chebyshev's inequality Could you have a distribution with $99$ values of a little less than $23$ and one value a little greater than $35$? What is the standard deviation of that?