Statistics, can anyone help me?

Question

Consider the data set that is summarized in Minitab Output below:
Stem-and-leaf of Data
N = 30
Leaf Unit = 1.0

4 2 3469
7 3 178
10 4 157
15 5 36679
15 6 124566799
6 7 36
4 8 34
2 9 26


N= 30
N*=0
Mean=57.17
SE Mean=3.62
StDev=19.84
Minimum=23
Q1=40.25
Median=60
Q3=69
Maximum=96

Answer the following:
(a) According to Chebyshev’s rule what is the minimum number of observations that will fall between 17.49 and 96.85 ?

( b) What is actual number of observations that fall in the interval in (a)?

Answer 1

I don't know how to interpret Minitab data like the above; I only see 24 values up there, not 30.

In any case, since mean=57.17 and stdev=19.84, the minimum number of observations between 17.49 and 96.85 can be found this way: 96.85-57.17=57.57-17.49=39.68, and 39.68/19.84=2, so by Chebyshev's inequality, which states that

P(|X-mean| >= k*stdev) <=1/k^2,

you have k=2, because you want |X-mean|>=39.68, so

P(|X-57.17| >= 39.68) <= 1/4.

Therefore, the probability that an observation is between 17.49 and 96.85 is at least 1-1/4=3/4. Therefore, the minimum number is ceiling((3/4)*30) = 23.