## Answers

Given that about 36% of all U.S. adults will try to pad their insurance claims.

Hence p=probability that an U.S. adult will try to pad their insurance claims=0.36

Now the office received 124 insurance claims to be processed in the next few days.

Let X=Number of claims that are being padded.

What is Binomial Distribution?

A discrete random variable X is said to have a binomial distribution if its PMF(Probability Mass Function) is given by,

Notation: X~Binomial(n,p)

Now here n is very large that makes our calculations hard and time consuming that is why we use normal approximation to the binomial distribution.

We can use normal approximation if np and np(1-p) are both greater than 5

Hence we can use normal approximation to binomial.

Normal Approximation to the Binomial Distribution

If X~Binomial(n,p) then for large n,

[Approximately for large n]

**Normal Distribution**

A continuous random variable X is said to have a normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Notation:

**Standard Normal Distribution**

A continuous random variable X is said to have a standard normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

**Exact evaluation of** ?(x) is not possible but numerical method can be applied. The values of ?(x) has been tabulated extensively in Biometrika Volume I.

Notation:

Continuity Correction Factor

Continuity correction is a correction that we use when a continuous distribution is used to approximate a discrete distribution.

Coming back to our problem,.

(a) Half or more of the plans have been padded,

Applying continuity correction,

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

Applying continuity correction,

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

Applying continuity correction,

(d) More than 80 of the claims have been padded,

Applying continuity correction,

~ Normal(0,1) Vnp1 - p)