Let the n-th observation be xn.
Given mean xˉ=8, we have:
n∑i=1n−1xi+xn=8
n48+xn=8⇒xn=8n−48
Given variance σ2=16, we have:
n∑i=1n−1xi2+xn2−(xˉ)2=16
n496+xn2−64=16
n496+xn2=80
496+xn2=80n
Substituting xn=8n−48:
496+(8n−48)2=80n
496+64n2−768n+2304=80n
64n2−848n+2800=0
Dividing by 16:
4n2−53n+175=0
4n2−28n−25n+175=0
4n(n−7)−25(n−7)=0
(4n−25)(n−7)=0
Since n must be an integer, n=7.
Answer: 7