Need help In the following problem, check that it is appropriate to use the normal appreciation...

Question

need help

In the following problem, check that it is appropriate to use the normal appreciation to the binomial. Then use the normal distribution to estimate the requested probabilities Do you try to padan insurance claim to cover your deductible? About 36% of all U.S. adults will try to pad their insurance dans Suppose that you are the director of an insurance adjustment office. Your office has just received 124 Insurance aims 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 dams have been added (b) fewer than 45 of the claims have been padded (c) from 40 to 64 of the claims have been padded (d) more than 10 of the claims have not been added

 · Answer

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.

$X ~ Binomial(n = 124, p=0.36)$

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,

$fa(z) = {(%)*(1 – p)** , 10 = 0,1,2,., , other wise$

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

$np = 124 * 0.36 = 44.64 > 5$

$np(1 – 2) = 124 * 0.36 * 0.64 = 28.57 = 44.64 > 5$

Hence we can use normal approximation to binomial.

Normal Approximation to the Binomial Distribution

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

$Z=- 1 - Tp. ~ Normal(0,1) Vnp1 - p)$ [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

$fxle) **),ch$

its CDF(Cumulative Distribution Function) is given by,

$P(X\leq x)=F(x)=\int_{-\infty }^{x}f_{X}(t)dt=\int_{-\infty }^{x}\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left ( \frac{t-\mu }{\sigma } \right )^2}dt$

Notation:

$X Normal(u.o$

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

$-e-reR 2T fx(x)$

its CDF(Cumulative Distribution Function) is given by,

$P(X <) = (2) = fx(t)dt = -e #² at J- 0 27T$

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:

$XNormal (0, 1)$

Continuity Correction Factor

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

$P(X = a) = Pla -0.5 < X <a +0.5)$

Coming back to our problem,.

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

$P\left ( X\geq \frac{124}{2} \right )$

$= P(X > 62)$

$= 1- P(X < 62)$

Applying continuity correction,

$=1-P(X<61.5)$

$=1-P\left ( \frac{X-np}{\sqrt{np(1-p)}}<\frac{61.5-np}{\sqrt{np(1-p)}} \right )$

$=1-P\left ( Z<\frac{61.5-np}{\sqrt{np(1-p)}} \right )[Z\sim Normal(0,1)]$

$61.5 – 44.64 =1-PZ < 28.5696$

$=1-P\left ( Z<3.15 \right )$

$=1-\Phi (3.15)$

$=1-0.9992$

$=0.0008$

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

$P(X <45)$

Applying continuity correction,

$=P\left ( X< 44.5 \right )$

$=P\left ( \frac{X-np}{\sqrt{np(1-p)}}<\frac{44.5-np}{\sqrt{np(1-p)}} \right )$

$=P\left ( Z<\frac{44.5-np}{\sqrt{np(1-p)}} \right )[Z\sim Normal(0,1)]$

$=P\left ( Z<\frac{44.5-44.64}{\sqrt{28.5696}} \right )$

$=P\left ( Z<-0.03 \right )$

$=\Phi (-0.03)$

$=1 – Φ(0.03)$

$=1-0.5120=0.488$

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

$P(40\leq X\leq 64)$

Applying continuity correction,

$=P(39.5< X< 64.5)$

$=P(X<64.5)-P(X<39.5)$

$=P\left ( \frac{X-np}{\sqrt{np(1-p)}}<\frac{64.5-np}{\sqrt{np(1-p)}} \right )-P\left ( \frac{X-np}{\sqrt{np(1-p)}}<\frac{39.5-np}{\sqrt{np(1-p)}} \right )$

$=P\left ( Z<\frac{64.5-np}{\sqrt{np(1-p)}} \right )-P\left ( Z<\frac{39.5-np}{\sqrt{np(1-p)}} \right )[Z~Normal(0,1)]$

$=P\left ( Z<\frac{64.5-44.64}{\sqrt{28.5696}} \right )-P\left ( Z<\frac{39.5-44.64}{\sqrt{28.5696}} \right )$

$=P\left ( Z<3.72 \right )-P\left ( Z<-0.96 \right )$

$=\Phi \left ( 3.72 \right )-\Phi \left ( -0.96 \right )$

$= Φ(3.72) - (1 – Φ(0.96))$

$=0.9999-(1-0.8315)$

$=0.9999-0.1685$

$=0.8314$

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

$P(X>80)$

$=1-P(X\leq 80)$

Applying continuity correction,

$= 1- P(X < 80.5)$

$= 1-PL X - np np(1- p 80.5 – np np(1-P)$

$=1-P\left ( Z<\frac{80.5-np}{\sqrt{np(1-p)}} \right )[Z\sim Normal(0,1)]$

$=1-P\left ( Z<\frac{80.5-44.64}{\sqrt{28.5696}} \right )$

$=1-P\left ( Z<6.71 \right )$

$=1 – Φ(6.71)$

$\simeq 1-1\simeq 0$

.

Need help In the following problem, check that it is appropriate to use the normal appreciation...

Question:

Answers

Similar Solved Questions

13. For each of the following reactions show the reactive electrophile in the EAS reaction and...

Mary has come to your clinic. She is a single mom with 3 children. She is...

Question 2: Ronald’s doctor has given him a prescription for a pharmaceutical grade “fish-oil” supplement. Explain...

17. Which of the following represents normal cost measurement a. Actual Direct Materials + Actual Direct...

What are four ways in which gene expression is regulated? eukaryotes

-1 -1 -1 yIn] -1 LTI systemas shown in the 4. Given a causal signal flow graph above where the coefficient r is real: (a) Determine the system function, H(z).(5) (b) Determine a minimummultiply I...

Calculate a) Req b) The current through each resistor c) The total power d) The power...

In the data below the rows define population groups A,B,C,D and the columns are education levels...

Rep: ID: Consider a box with two chambers labeled 1 and 2. Each chamber is filled...

What is your reaction to the idea of there being a climate debt owed by richer...

Consider the following A9 -24 (a) Verify that A is diagonalizable by computing P AP (b)...

M cmo MGT120 2020-02-02 Use the following information to answer questions 13-15 inclusive. The income statement...

Luzadis Company makes furniture using the latest automated technology. The company uses a job-order costing system...

An object weighs 10 N. You need to push down with 30 N in order to...

Automata Prove that regular languages are closed under difference, using an indirect proof (leveraging the closure...

How are fine motor skills defined? What activities prepare children for writing and drawing? How does...

For the function shown in the graph, find the indicated function value and limit. Find f(2)...

5. Consider a reaction starting with reactants and products in nonstandard state conditions, and a negative...

Question Help The net income of a company for the year ended was 5500,000. The company...