In Exercises 9–14, 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

2:07 minutes

Problem 5.5.13

Textbook Question

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x ≤ 150)

Verified step by step guidance

Step 1: Understand the problem. The given problem involves a binomial probability, P(x ≤ 150), which means we are looking for the probability that the random variable x is less than or equal to 150 in a binomial distribution.

Step 2: Recall the conditions for using a normal approximation to a binomial distribution. The binomial distribution can be approximated by a normal distribution if the sample size is large enough, specifically if both np ≥ 5 and n(1-p) ≥ 5, where n is the number of trials and p is the probability of success.

Step 3: Apply the continuity correction. Since the binomial distribution is discrete and the normal distribution is continuous, we use a continuity correction. For P(x ≤ 150), we adjust the value to P(x ≤ 150.5) to account for the discrete-to-continuous transition.

Step 4: Standardize the value using the z-score formula. The z-score formula is z = (x - μ) / σ, where μ = np (mean of the binomial distribution) and σ = √(np(1-p)) (standard deviation of the binomial distribution). Substitute the values of n, p, and x = 150.5 into the formula to calculate the z-score.

Step 5: Use the standard normal distribution table or a statistical software to find the probability corresponding to the calculated z-score. This will give you the approximate probability for P(x ≤ 150) using the normal distribution with continuity correction.

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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. This concept is essential for understanding discrete outcomes in scenarios like coin flips or quality control in manufacturing.

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 approximate a normal distribution due to the Central Limit Theorem, which states that the sum of a large number of independent random variables will be normally distributed, 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. It involves adjusting the discrete value by 0.5 units to account for the fact that the normal distribution is continuous. This correction improves the accuracy of the approximation, especially when the number of trials is small or the probability of success is not extreme.

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In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.P(x ≤ 150)