In Exercises 63–68, write the binomial probability in words. Then...

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

2:10 minutes

Problem 5.R.64

Textbook Question

In Exercises 63–68, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x ≤ 36)

Verified step by step guidance

Step 1: Interpret the binomial probability in words. The problem asks for the probability of obtaining 36 or fewer successes in a binomial experiment.

Step 2: Recall that a binomial distribution can be approximated by a normal distribution when the sample size is large and both np and n(1-p) are greater than or equal to 5. Verify these conditions for the given problem.

Step 3: Identify the mean (μ) and standard deviation (σ) of the binomial distribution. Use the formulas μ = n * p and σ = sqrt(n * p * (1 - p)), where n is the number of trials and p is the probability of success.

Step 4: Apply the continuity correction. Since the binomial probability is P(x ≤ 36), convert it to the normal distribution probability P(x ≤ 36.5) to account for the discrete-to-continuous adjustment.

Step 5: Standardize the value 36.5 using the z-score formula z = (x - μ) / σ, where x is the value of interest, μ is the mean, and σ is the standard deviation. Then, use the standard normal distribution table or a calculator to find the corresponding probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Probability

Binomial probability refers to the likelihood of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is calculated using the binomial formula, which incorporates the number of trials, the number of successes, and the probability of success on each trial. This concept is essential for understanding discrete outcomes in scenarios like coin flips or quality control.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is significant in statistics because many phenomena tend to follow this distribution due to the Central Limit Theorem, which states that the sum of a large number of independent random variables will approximate a normal distribution, regardless of the original distribution.

Continuity Correction

Continuity correction is a technique used when approximating a discrete probability distribution, like the binomial distribution, with a continuous distribution, such as the normal distribution. This correction involves adjusting the discrete value by 0.5 in either direction to better align the probabilities. For example, when calculating P(x ≤ 36) in a binomial context, one would use P(x ≤ 36.5) in the normal approximation to account for the discrete nature of the binomial variable.

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In Exercises 63–68, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.P(x ≤ 36)