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Chapter 6: Problem 70

Find the following binomial probabilities using the normal approximation. a. \(n=140, \quad p=.45, \quad P(x=67)\) b. \(n=100, \quad p=.55, \quad P(52 \leq x \leq 60)\) c. \(n=90, \quad p=.42, \quad P(x \geq 40)\) d. \(n=104, \quad p=.75, \quad P(x \leq 72)\)

Short Answer

Expert verified
a. The probability that x=67 is approximately 0.056. b. The probability that 52 <= x <= 60 is approximately 0.47. c. The probability that x >= 40 is approximately 0.64. d. The probability that x <= 72 is approximately 0.029.

Step by step solution

01

Check the validity of Normal Approximation

Verify that \(n*p\) and \(n*(1-p)\) are both greater than 5 for each part. If yes, then the normal approximation is valid.
02

Calculation of Mean and Standard Deviation

Calculate the mean (\(渭 = n*p\)) and the standard deviation (\(蟽 = sqrt(n*p*(1-p))\)) for each part.
03

Calculation of Z-Score

Convert x to z using the formula \(z = (x - 渭) / 蟽\) for each part. If there is a range from x1 to x2, calculate two z-scores (\(z1 = (x1 - 渭) / 蟽\) and \(z2 = (x2 - 渭) / 蟽\)).
04

Finding the Probability

For each part, use the z-scores to find the probabilities. Use a standard normal distribution table to find the probabilities corresponding to the z-score. If there is a range from z1 to z2, the probability \(P(z1 鈮� z 鈮� z2) = P(z2) - P(z1)\). If there is a condition like \(x 鈮� x1\) or \(x 鈮� x1\), use z = \(z1\) in \(P(z 鈮� z1)\) or \(P(z 鈮� z1)\), respectively.

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