Textbook Question
Give three interpretations for the area under a normal curve.
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6. Normal Distribution and Continuous Random Variables
Non-Standard Normal Distribution
1:42 minutesProblem 7.4.1
Textbook Question
In a binomial experiment with n trials and probability of success p, if __ ________, the binomial random variable X is approximately normal with μX = ________ and σX = ________.
Verified step by step guidance1
Identify the conditions under which a binomial distribution can be approximated by a normal distribution. Typically, this approximation is valid when both the expected number of successes and failures are sufficiently large, often stated as \(n p \geq 10\) and \(n (1 - p) \geq 10\).
Recall the mean (expected value) of a binomial random variable \(X\) with parameters \(n\) and \(p\) is given by the formula: \(\mu_X = n p\).
Recall the standard deviation of the binomial random variable \(X\) is given by the formula: \(\sigma_X = \sqrt{n p (1 - p)}\).
Understand that when the conditions for normal approximation are met, the binomial distribution \(X\) can be approximated by a normal distribution with mean \(\mu_X\) and standard deviation \(\sigma_X\) as calculated above.
Summarize the result: If \(n p \geq 10\) and \(n (1 - p) \geq 10\), then \(X \sim \text{Binomial}(n, p)\) is approximately \(N(\mu_X, \sigma_X^2)\) where \(\mu_X = n p\) and \(\sigma_X = \sqrt{n p (1 - p)}\).
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. It is discrete and defined by parameters n (number of trials) and p (probability of success).
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Mean & Standard Deviation of Binomial Distribution
Normal Approximation to the Binomial
When the number of trials n is large and both np and n(1-p) are greater than or equal to 10, the binomial distribution can be approximated by a normal distribution. This simplifies calculations by using continuous probability methods.
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Mean and Standard Deviation of Binomial Distribution
The mean (μX) of a binomial random variable X is given by np, representing the expected number of successes. The standard deviation (σX) is √(np(1-p)), measuring the variability around the mean.
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