Identifying if binomial probabilities can be approximated by the normal distribu

Question

Identifying if binomial probabilities can be approximated by the normal distribution A study has shown that 93% of Internet users have shopped online ( Source Pew internet and American Life project (2008) . Report on online Activities and Pursuits, February 13.) Consider this binomial probability: in a random sample of 60 internet users, what is the probability that fewer than 50 shop online? It____________appropriate to use the normal distribution to approximate the binomial probability, because ______________ (Hint: Note that N is the sample size, P is the population proportion, and q=1-p )

Explanation / Answer

Solution

Back-up Theory

       If X ~ B(n, p), np 5 and np(1 - p) 5 then Binomial probabilities can approximated by

       Normal by Z = (X – np)/ {np(1 - p)} ~ N(0, 1)

Now, to work out solution,

We have n = 60, p = 0.93, and hence 1 – p = 0.07.

While np = 55.8 > 5, np(1 - p) < 5. So, it is not advisable to employ Normal Approximation to Binomial.

Using Excel Function for Binomial Distribution, probability of fewer than 50 for B(60, 0.93) is 0.002749.

Normal approximation puts the probability as 0.00029.

DONE

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