Question

If possible, please explain how to calculate with a Ti-84 Plus calculator.

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Do you try to pad an insurance claim to cover your deductible? About 44% of all U.S. adults will try to pad their insurance claims! Suppose that you are the director of an insurance adjustment office. Your office has just received 122 insurance claims to be processed in the next few days. Find the following probabilities. (Round your answers to four decimal places.)

(a) half or more of the claims have been padded


(b) fewer than 45 of the claims have been padded


(c) from 40 to 64 of the claims have been padded


(d) more than 80 of the claims have not been padded

Answer 1

(a)

n = 122

p = 0.44

q = 1 - p = 0.56

= np = 122 X 0.44 = 53.68

To find P(X61):

Applying Continuity Correction:
To find P(X>60.5):

Z= (60.5 - 53.68)/5.4828 = 1.24

Table of Area Under Standard Normal Curve gives area = 0.3925

So

P(X>61) = 0.5 - 0.3925 = 0.1075

So,

Answer is:

0.1075

(b)

n = 122

p = 0.44

q = 1 - p = 0.56

= np = 122 X 0.44 = 53.68

To find P(X<45):

Applying Continuity Correction:
To find P(X<44.5):

Z= (44.5 - 53.68)/5.4828 = - 1.67

Table of Area Under Standard Normal Curve gives area = 0.4525

So

P(X<45) = 0.5 - 0.4525 = 0.0475

So,

Answer is:

0.0475

(c)

n = 122

p = 0.44

q = 1 - p = 0.56

= np = 122 X 0.44 = 53.68

To find P(40<X<64):

Applying Continuity Correction:
To find P(39.5<X<64.5):

Case 1: For X from 39.5 to mid value:

Z= (39.5 - 53.68)/5.4828 = - 2.59

Table of Area Under Standard Normal Curve gives area = 0.4952

Case 2: For X from mid value to 64.5

Z= (64.5 - 53.68)/5.4828 = 1.97

Table of Area Under Standard Normal Curve gives area = 0.4756

So

P(40<X<64) = 0.4952 + 0.4756 = 0.9708

So,

Answer is:

0.9708

(e)

More than 80 not padded is same as less than or equal to 122 - 80 = 42 is padded

n = 122

p = 0.44

q = 1 - p = 0.56

= np = 122 X 0.44 = 53.68

To find P(X42):

Applying Continuity Correction:
To find P(X<42.5):

Z= (42.5 - 53.68)/5.4828 = - 2.04

Table of Area Under Standard Normal Curve gives area = 0.4793

So

P(X42) = 0.5 - 0.4793 = 0.0207

So,

Answer is:

0.0207