Question

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 36% 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 .9996 Incorrect: Your answer is incorrect. (b) fewer than 45 of the claims have been padded 0.5739 Incorrect: Your answer is incorrect. (c) from 40 to 64 of the claims have been padded 0.2703 Incorrect: Your answer is incorrect. (d) more than 80 of the claims have not been padded 1.0000 your answer is incorrect

Answer 1

Part a)

Mean = n * P = ( 122 * 0.36 ) = 43.92
Variance = n * P * Q = ( 122 * 0.36 * 0.64 ) = 28.1088
Standard deviation = = 5.3018

P ( X >= 61 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 61 - 0.5 ) =P ( X > 60.5 )

P ( X > 60.5 ) = 1 - P ( X < 60.5 )
Standardizing the value

Z = ( 60.5 - 43.92 ) / 5.3018
Z = 3.13

P ( Z > 3.13 )
P ( X > 60.5 ) = 1 - P ( Z < 3.13 )
P ( X > 60.5 ) = 1 - 0.9991
P ( X > 60.5 ) = 0.0009

Part 2)

P ( X < 45 )
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 45 - 0.5 ) = P ( X < 44.5 )

P ( X < 44.5 )
Standardizing the value

Z = ( 44.5 - 43.92 ) / 5.3018
Z = 0.11

P ( X < 44.5 ) = P ( Z < 0.11 )
P ( X < 44.5 ) = 0.5438

Part c)

P ( 40 <= X <= 64 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 40 - 0.5 < X < 64 + 0.5 ) = P ( 39.5 < X < 64.5 )

P ( 39.5 < X < 64.5 )
Standardizing the value

Z = ( 39.5 - 43.92 ) / 5.3018
Z = -0.83
Z = ( 64.5 - 43.92 ) / 5.3018
Z = 3.88
P ( -0.83 < Z < 3.88 )
P ( 39.5 < X < 64.5 ) = P ( Z < 3.88 ) - P ( Z < -0.83 )
P ( 39.5 < X < 64.5 ) = 0.9999 - 0.2022
P ( 39.5 < X < 64.5 ) = 0.7977

Part d)

P ( X > 80 )
Using continuity correction
P ( X > n + 0.5 ) = P ( X > 80 + 0.5 ) = P ( X > 80.5 )

P ( X > 80.5 ) = 1 - P ( X < 80.5 )
Standardizing the value

Z = ( 80.5 - 43.92 ) / 5.3018
Z = 6.9

P ( Z > 6.9 )
P ( X > 80.5 ) = 1 - P ( Z < 6.9 )
P ( X > 80.5 ) = 1 - 1
P ( X > 80.5 ) = 0