Question

Decide 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 found that 79% of those who text on cell phones receive spam or unwanted messages. You randomly select 100 adults who text on cell phones. Complete parts (a) through (d).

Determine whether a normal distribution can be used to
approximate the binomial distribution. Choose the correct answer
below.

A. Yes, because both npgreater than or equals5 and nqgreater than
or equals5.

Find the probability that the number of people who receive spam or unwanted messages is exactly 83.

Answer #1

Let X denotes the number of adults who text on cell phones receive spam or unwanted messages out of 100 randomly selected adults.

X ~ Binomial(n = 100, p = 0.79)

Now,

E(X) = 100*0.79 = 79

Var(X) = 100*0.79*(1 - 0.79) = 16.59

and

Using normal approximation to binomial, we get

Now,

The probability that the number of people who receive spam or unwanted messages is exactly 83

(using minitab software, calc -> probability distributions -> normal)

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np, nq formula
A) Explain when you can use normal distribution to approximate
the binomial distribution
B) If we already know the binomial probability distribution
formula, why do we need to know this one?
C)When is it useful to approximate the binomial distribution as
normal. Provide 1 example

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Explain how to decide when a normal distribution can be used to
approximate a binomial distribution.
Choose the correct answer below.
A.
If
npgreater than or equals?5
and
nqgreater than or equals??5,
the normal distribution can be used.
B.
If
npnot equals?5
and
nqnot equals??5,
the normal distribution can be used.
C.
If
npless than<5
and
nqless than<?5,
the normal distribution can be used.
D.
If
npgreater than>0
and
nqgreater than>?0,
the normal distribution can be used.

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Q: By how much do the exact and approximated probabilities
differ?
A. ____(Round to four decimal places as needed.)
B. The normal distribution cannot be used.

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