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 269 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 ≥ 5 and nq ≥ 5.

b.​No, because np < 5.

c.​No, because nq <5.

A) ​- Find the probability that exactly 20 workers will say yes.

What is the indicated​ probability? (____) 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 7 workers will say yes.

What is the indicated​ probability? (____) Round to four decimal places as​ needed.

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

C) - Find the probability that fewer than 20 workers will say yes.

What is the indicated​ probability? (____) Round to four decimal places as​ needed.

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

D) - A transit authority offers discount rates to companies that have at least 30 employees who use public transportation to get to work. There are 452 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 nq < 5.

b.​ No, because np < 5.

c.​ Yes, because both np ≥ 5 and nq ≥ 5.

What is the probability that the company will not get the​ discount? (____) Round to four decimal places as​ needed.

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

Answer 1

Here we have

n= 269, p = 0.05

Since np and n(1-p) are greater than 5 so normal approximation is appropriate.

Correct option is:

a.​Yes, because both np ≥ 5 and nq ≥ 5.

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Let X is a random variable say yes. Here X has approximately normal distribution with mean

and standard deviation is

(a)

The z-score for X = 20-0.5 = 19.5 is

The z-score for X = 20+0.5 = 20.5 is

The probability that exactly 20 workers will say yes is

Excel function used: "=NORMSDIST(1.97)-NORMSDIST(1.69)"

Following is the curve:

(b)

The z-score for X = 7 - 0.5 = 6.5 is

So,

Excel function used: "=1-NORMSDIST(-1.94)"

Following is the graph:

(c)

The z-score for X = 20 - 0.5 = 19.5 is

So,

Excel function used: "=NORMSDIST(1.69)"

Following is the graph:

(d)

Here we have n=452, p= 0.05

Here X has approximately normal distribution with mean

and standard deviation is

The z-score for X = 30 - 0.5 = 29.5 is

So,

Excel function used: "=1-NORMSDIST(1.49)"

Following is the graph: