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. Five percent of workers in a city use public transportation to get to work. You randomly select 254 workers and ask them if they use public transportation to get to work. Complete parts (a) through (d).

Can the normal distribution be used to approximate the binomial distribution?

A.

Yes, because both

np greater than or equals≥5

and

nq greater than or equals≥5.

B.

No, because

nq less than<5.

C.

No, because

np less than<5.

(a) Find the probability that exactly

1919

workers will say yes.

What is the indicated probability?

nothing

(Round to four decimal places as needed.)

Sketch the graph of the normal distribution with the indicated probability shaded.

(b) Find the probability that at least

77

workers will say yes.

What is the indicated probability?

nothing

(Round to four decimal places as needed.)

Sketch the graph of the normal distribution with the indicated probability shaded.

A.

1317x

mu equals 12.7μ=12.7

x y graph

B.

1317x

mu equals 12.7μ=12.7

x y graph

C.

1317x

mu equals 12.7μ=12.7

x y graph

D.

The normal distribution cannot be used to approximate the binomial distribution.

(c) Find the probability that fewer than

1919

workers will say yes.

What is the indicated probability?

nothing

(Round to four decimal places as needed.)

Sketch the graph of the normal distribution with the indicated probability shaded.

A.

13119x

mu equals 12.7μ=12.7

x y graph

B.

13119x

mu equals 12.7μ=12.7

x y graph

C.

13119x

mu equals 12.7μ=12.7

x y graph

D.

The normal distribution cannot be used to approximate the binomial distribution.

(d) A transit authority offers discount rates to companies that have at least

3030

employees who use public transportation to get to work. There are

499499

employees in a company. What is the probability that the company will not get the discount?

Can the normal distribution be used to approximate the binomial distribution?

A.

No, because

npless than<5.

B.

Yes, because both

npgreater than or equals≥5

and

nqgreater than or equals≥5.

C.

No, because

nqless than<5.

What is the probability that the company will not get the discount?

nothing

(Round to four decimal places as needed.)

Sketch the graph of the normal distribution with the indicated probability shaded.

Answer 1

The number of workers out of who use public transportation to get to work follows binomial distribution with parameters . Where is the proportion of workers who uses public transportation.

The PMF of is  .

Here and

We can approximate the binomial distribution by an equivalent Normal distribution if

And the normal distribution has mean and variance .

Since , correct choice is

A. Yes, because both np greater than or equals ≥5 and nq greater than or equals ≥5.

The probability that exactly 19 workers will say Yes is

The R command for the above probability is

> dbinom(19,254,0.05)
[1] 0.02251518

The probability that at least 7 workers will say Yes is

The normal approximation is (Here

The region is shown shaded.

c) The probability that fewer than 19 workers will say Yes is

The norma approximation is

The region is shown shaded.

We are required to solve only 4 parts. Please post the remaining questions as another post.
We do not get any additional amount for solving more.