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

52

​%

have encountered fraudulent charges on their credit cards. You randomly select

100

adults in the region. Complete parts​ (a) through​ (d) below.

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 equals

5

and

nqgreater than or equals

5.

B.

​No, because

nqless than

5.

C.

​No, because

npless than

5.

​(a) Find the probability that the number who have encountered fraudulent charges on their credit cards is​ (a) exactly

55

​,

​(b) at least

55

​,

and​ (c) fewer than

55

.

nothing

​(Round to four decimal places as​ needed.)

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

A.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52, 54.5, and 55.5. The area under the curve between 54.5 and 55.5 is shaded.

B.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52 and 54.5. The area under the curve to the right of 54.5 is shaded.

C.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52 and 54.5. The area under the curve to the left of 54.5 is shaded.

D.

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

​(b) Find the probability that the number who have encountered fraudulent charges on their credit cards is at least

55

.

nothing

​(Round to four decimal places as​ needed.)

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

A.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52, 54.5, and 55.5. The area under the curve between 54.5 and 55.5 is shaded.

B.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52 and 54.5. The area under the curve to the right of 54.5 is shaded.

C.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52 and 54.5. The area under the curve to the left of 54.5 is shaded.

D.

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

​(c) Find the probability that the number who have encountered fraudulent charges on their credit cards is fewer than

55

.

nothing

​(Round to four decimal places as​ needed.)

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

A.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52, 54.5, and 55.5. The area under the curve between 54.5 and 55.5 is shaded.

B.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52 and 54.5. The area under the curve to the left of 54.5 is shaded.

C.

553569x

mu equals 52

A normal curve is over a horizontal axis and is centered on 52. Vertical line segments extend from the horizontal axis to the curve at 52 and 54.5. The area under the curve to the right of 54.5 is shaded.

D.

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

​(d) Identify any unusual events. Explain.

A.

The event in part​ (c) is unusual because its probability is less than 0.05.

B.

The event in part​ (a) is unusual because its probability is less than 0.05.

C.

The event in part​ (b) is unusual because its probability is less than 0.05.

D.

There are no unusual​ events, because all of the probabilities are greater than 0.05.

Answer 1

n = 100

p = 0.52

np = 100 * 0.52 = 52

nq = n(1 - p) = 100(1 - 0.52) = 48

Option - A) Yes, because both np greater than or equals to 5 and nq greater than or equals 5.

a) = np = 100 * 0.52 = 52

   = sqrt(np(1 - p))

       = sqrt(100 * 0.52 * 1 - 0.52))

       = 7.175

P(X = 55)

= P((54.5 - )/< (X - )/< (55.5 - )/)

= P((54.5 - 52)/7.175 < Z < (55.5 - 52)/7.175)

= P(0.348 < Z < 0.488)

= P(Z < 0.488) - P(Z < 0.348)

= 0.6872 - 0.6361

= 0.0511

b) P(X > 55)

= P((X - )/> (54.5 - )/)

= P(Z > (54.5 - 52)/7.175)

= P(Z > 0.348)

= 1 - P(Z < 0.348)

= 1 - 0.6361

= 0.3639

c) P(X < 55)

= P(X < 54)

= P((X - )/< (54.5 - )/)

= P(Z < (54.5 - 52)/7.175)

= P(Z < 0.348)

= 0.6361

d) Option - D) There are no unusual events, because all of the probabilities are greater than 0.05