Chapter 8: Q96E (page 452)

Assume that x is a binomial random variable with n = 1000 andp = 0.50. Use a normal approximation to find each of the following probabilities:
a. $P (x > 500)$
b. $P (490 \leq x < 500)$
c. $P (x > 550)$

Short Answer

Expert verified

$P (x > 500) = 0.50$
$P (490 \leq x \leq 500) = 0.2357$

c. $P (x > 550) = 0.0008$

Step by step solution

Given information

x be binomial distribution with n = 1000 and p= 0.50

Calculation for the z value

$\begin{array}{l} μ = n p \\ = 1000 \times 0.50 \\ = 500 \end{array}$

$σ = \sqrt{n p (1 - p)} = \sqrt{1000 \times 0.50 \times (1 - 0.50)} = \sqrt{1000 \times 0.50 \times 0.50} = 15.8114$

So,

$\begin{array}{l} z = \frac{x - μ}{σ} \\ = \frac{x - 500}{15.8114} \end{array}$

Probability calculation P(x>500)

$P (x > 500) = 1 - P (x < 500) = 1 - P (z < 0) = 1 - 0.50 = 0.50 P (x > 500) = 0.50$

Therefore, the required probability is 0.50.

Probability calculation P(490≤x<500)

$P (490 \leq x < 500) = P (< 500) - P (x \leq 490) = P (z < 0) - P (z \leq - 0.6325) = P (z < 0) - (1 - P (z \leq - 0.6325)) = 0.50 - (1 - 0.7357) = 0.2357 P (490 \leq x < 500) = 0.2357$

Therefore, the probability is 0.2357.

Probability calculation P(x>550)

$P (x > 550) = 1 - P (x > 550) = 1 - P (z < 3.1623) = 1 - 0.999217 = 0.000783 \approx 0.0008 P (x > 550) = 0.0008$

Therefore, the probability is 0.0008.

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Sample 1	Sample 2
$\begin{array}{l} n_{1} =58 \\ {\bar{x}}_{1} =17.5 \end{array}$	$\begin{array}{l} n_{2} =62 \\ {\bar{x}}_{2} =16.23 \end{array}$

Assume that x is a binomial random variable with n = 1000 andp = 0.50. Use a normal approximation to find each of the following probabilities:
a. $P (x > 500)$
b. $P (490 \leq x < 500)$
c. $P (x > 550)$

Short Answer

Step by step solution

Given information

Calculation for the z value

Probability calculation P(x>500)

Probability calculation P(490≤x<500)

Probability calculation P(x>550)

One App. One Place for Learning.

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Probability and Statistics

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Assume that x is a binomial random variable with n = 1000 andp = 0.50. Use a normal approximation to find each of the following probabilities:a. P(x>500)b.P(490≤x<500)c.P(x>550)

Short Answer

Step by step solution

Given information

Calculation for the z value

Probability calculation P(x>500)

Probability calculation P(490≤x<500)

Probability calculation P(x>550)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Pure Maths

Geometry

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Assume that x is a binomial random variable with n = 1000 andp = 0.50. Use a normal approximation to find each of the following probabilities:
a. $P (x > 500)$
b. $P (490 \leq x < 500)$
c. $P (x > 550)$