Peanut and tree nut allergies are consideredto be the most serious food allerigi

Question

Peanut and tree nut allergies are consideredto be the most serious food allerigies. According to the national institue of allergy and infectious diseases, roughly 1% of Americans are allergic to peanuts or tree nuts. A random sample of 1500 americans is obtained.

a) Explain why a large sample size is needed to for the distribution of the sample proportion to be approximately normal?

b) Would it be unusual if a random sample of 1500 americans results in a fewer than 10 with peanut or tree nuts allergies?

Explanation / Answer

if a sample size is less than 5% of the population size and the value np*(1-p) >10 , we find that the sample size n such that the condition np*(1-p) >= 10

here p = 0.01
so

n*0.01*0.99 >=10
this becomes
n*0.0099 >= 10
hence N must be sufficiently large to satisfy the above inequality

atleast 1010.10 in thsi case

b) we can find the 95% confidence interval of the proportion as

p +- z*sqrt((p*1-p)/n) , here n = 1500 and z = 1.96 for 95% CI

0.01 +- 1.96*sqrt(0.01*0.99/1500)

0.005 and 0.15
between .5% and 1.5% it is usual

so if 10/1500 = 0.006 which is 0.6%
hence it falls within the 95% CI range , hence it is not unusual

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