Question

Determine whether you can use the normal distribution to approximate the binomial distribution. If you can, use the normal
distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use the
binomial distribution to find the indicated probabilities.
A survey of adults in a region found that 31% name professional football as their favorite sport. You randomly select 11
adults in the region and ask them to name their favorite sport. Complete parts (a) through (d) below
Determine whether a normal distribution can be used to approximate the binomial distribution. Choose the correct answer
below.

Answer 1

No normal distribution cannot be used to calculate the probability and and the probability is  0.02057

The sampling distribution of p-hat(sample proportion) is approximately normal with mean p and standard deviation √p(1-p)/n

It is known that, the conservative law of thumb for the sampling distribution of sample proportion to be approximately normal is,

np≥5

nq≥5

Here, n =11 , p = 0.31 and q = 0.69

Checking the condition for normality:

np = 11 x 0.31 = 3.41

3.41<5

Thus the condition fails

Thus, normal distribution is not appropriate.

Normal distribution is not appropriate because

The probability that the number who name professional football as their favorite sport is exactly 7 can obtained using binomial probability.

The required probability is calculated as follows:

P(X=7) = 11C$_{7}$ $(0.31)^7 (0.69)^4$

= 0.02057

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