6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:18 minutesProblem 5.R.61
Textbook Question
In Exercises 61 and 62, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.
A survey of U.S. adults ages 33 to 40 earning more than $150,000 per year found that 94% are content with how their lives have turned out so far. You randomly select 20 U.S. adults ages 33 to 40 earning more than $150,000 and ask if they are content with their lives so far.
Verified step by step guidance1
Step 1: Identify the parameters of the binomial distribution. In this problem, the number of trials (n) is 20, and the probability of success (p) is 0.94 (since 94% of the population is content).
Step 2: Check if the normal approximation to the binomial distribution can be used. The rule of thumb is that both np and n(1-p) must be greater than or equal to 5. Calculate np = 20 * 0.94 and n(1-p) = 20 * (1 - 0.94).
Step 3: If the conditions in Step 2 are satisfied, proceed to calculate the mean (μ) of the binomial distribution. The formula for the mean is μ = n * p.
Step 4: Calculate the standard deviation (σ) of the binomial distribution. The formula for the standard deviation is σ = sqrt(n * p * (1 - p)).
Step 5: If the conditions for normal approximation are not satisfied, explain that the binomial distribution cannot be approximated by a normal distribution in this case and provide reasoning based on the calculations in Step 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p). In this context, the success could be defined as an adult being content with their life.
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Normal Approximation to the Binomial
The normal approximation to the binomial distribution can be used when certain conditions are met, specifically when both np and n(1-p) are greater than or equal to 5. This allows for the use of the normal distribution to estimate probabilities and calculate the mean and standard deviation of the binomial distribution, simplifying analysis.
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Mean and Standard Deviation of a Binomial Distribution
The mean (μ) of a binomial distribution is calculated as μ = np, while the standard deviation (σ) is given by σ = √(np(1-p)). These formulas provide essential measures of central tendency and variability, helping to understand the distribution of successes in the context of the experiment.
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