Question

A newborn baby whose apgar score is over 6 is classified as normal and this happens in 80% of births. As a quality control check an auditor examines the records of 100 births. He would be suspicious if the number of normal births in the sample of 100 births fell above the upper limit of a 95% normal range. What is the upper limit?

Answer 1

Interesting problem.
Check if binomial theorem applies.
1. Is each trial a Bernoulli trial (either success or failure)? Yes
2. Is probability of success (Apgar><6) known and remain constant throughout experiment (audit)? Yes
3. Are trials (result of each birth) independent of each other? Basically yes, no indication to the contrary.
4. The number of trials is known and remains constant (100). Yes

Since all criteria are satisfied, we can use the binomial distribution to model the experiment, with which:
n=number of trials = 100
p=probability of success = 0.8
q=probability of failure = 1-0.8=0.2
μ=np = 100*0.8=80
σ²=npq=100*0.8*0.2=16
σ=√16=4

Since n>40, np>10, nq>10, we can approximate the distribution as a normal distribution (with appropriate continuity corrections).

The auditor is suspicious only if the numbers are too high, i.e. a upper-tail test.
So the limit is
μ+;1.644*σ
=86.58 Say 87.
So auditor would be suspicious if the number of normal births is above 87, for a sample of 100 births, at the 95% level.

Answer 2

So the limit is
μ+;1.644*σ
=86.58 Say 87.
So auditor would be suspicious if the number of normal births is above 87, for a sample of 100 births, at the 95% level
plz explain