SOLUTION: 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.

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Question 1200047: 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 37% 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 138 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 by textot(100) (Show Source):
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**1. Check for Normal Approximation**
* **n = 138 (sample size)**
* **p = 0.37 (probability of success - claim padded)**
* **q = 1 - p = 0.63 (probability of failure - claim not padded)**
* **np = 138 * 0.37 = 51.06 ≥ 5**
* **nq = 138 * 0.63 = 86.94 ≥ 5**
Since both np and nq are greater than 5, the normal approximation to the binomial distribution is appropriate.
**2. Calculate Mean (μ) and Standard Deviation (σ)**
* **μ = np = 138 * 0.37 = 51.06**
* **σ = √(npq) = √(138 * 0.37 * 0.63) = 5.67**
**3. Standardize and Use Standard Normal Table (or Calculator)**
* **(a) Half or more of the claims have been padded:**
* P(X ≥ 69)
* **Continuity Correction:** P(X ≥ 68.5)
* **Standardize:** z = (68.5 - 51.06) / 5.67 = 3.09
* **P(Z ≥ 3.09) = 0.0010** (using a standard normal table or calculator)
* **(b) Fewer than 45 of the claims have been padded:**
* P(X < 45)
* **Continuity Correction:** P(X < 44.5)
* **Standardize:** z = (44.5 - 51.06) / 5.67 = -1.17
* **P(Z < -1.17) = 0.1210**
* **(c) From 40 to 64 of the claims have been padded:**
* P(40 ≤ X ≤ 64)
* **Continuity Correction:** P(39.5 ≤ X ≤ 64.5)
* **Standardize:**
* z1 = (39.5 - 51.06) / 5.67 = -2.07
* z2 = (64.5 - 51.06) / 5.67 = 2.39
* **P(-2.07 ≤ Z ≤ 2.39) = 0.9817 - 0.0192 = 0.9625**
* **(d) More than 80 of the claims have not been padded:**
* If 80 claims have not been padded, then 58 claims have been padded.
* P(X > 58)
* **Continuity Correction:** P(X > 58.5)
* **Standardize:** z = (58.5 - 51.06) / 5.67 = 1.31
* **P(Z > 1.31) = 0.0951**
**Therefore:**
* **(a) Probability of half or more of the claims being padded: 0.0010**
* **(b) Probability of fewer than 45 claims being padded: 0.1210**
* **(c) Probability of from 40 to 64 claims being padded: 0.9625**
* **(d) Probability of more than 80 claims not being padded: 0.0951**