11 To apply Central Limit Theorem on sample proportions in O

#11 To apply Central Limit Theorem on sample proportions in One Sample Proportion test, the sample size and the population proportion under null hypothesis need to satisfy certain conditions. Which of the following scenarios meet the requirement? The sample size is 65 and the population proportion under null hypothesis is 88%. The sample size is 82 and the population proportion under null hypothesis is 80%. The sample size is 110 and the population proportion under null hypothesis is 5%. The sample size is 123 and the population proportion under null hypothesis is 7%.

Solution

When n (p) > 5 and n(1 - p ) > 5, the normal distribution is considered to be a good approximation to the binomial
distribution.
a.
n*p>5, 65*0.88> 5 => 57.2>5
n*(1-p)>5, 65*0.12> 5 => 57.2>5
Can apply Central Limit Theorem                  
                  
b.

n*p>5, 82*0.8> 5 => 65.6>5
n*(1-p)>5, 82*0.2> 5 => 65.6>5
Can apply Central Limit Theorem                  
                  
c.
n*p>5, 110*0.05> 5 => 5.5>5
n*(1-p)>5, 110*0.95> 5 => 5.5>5
Can apply Central Limit Theorem                  
                  
d.
n*p>5, 123*0.07> 5 => 8.61>5
n*(1-p)>5, 123*0.93> 5 => 8.61>5
Can apply Central Limit Theorem