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(55 < x < 60)

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Step 1: Understand the problem. The binomial probability P(55 < x < 60) represents the probability of the random variable x falling between 55 and 60 in a binomial distribution. The task is to convert this binomial probability into a normal distribution probability using a continuity correction.

Step 2: Recall the concept of continuity correction. Since the binomial distribution is discrete and the normal distribution is continuous, we apply a continuity correction by adjusting the boundaries of the interval. For P(55 < x < 60), the continuity correction expands the interval to P(54.5 < x < 60.5).

Step 3: Identify the parameters of the binomial distribution. The binomial distribution is defined by two parameters: n (number of trials) and p (probability of success). Ensure you know these values to proceed with the conversion to a normal distribution.

Step 4: Approximate the binomial distribution with a normal distribution. Use the formulas for the mean (μ = n * p) and standard deviation (σ = √(n * p * (1 - p)) of the binomial distribution to define the corresponding normal distribution.

Step 5: Convert the adjusted interval to a z-score. Using the normal distribution parameters (mean and standard deviation), calculate the z-scores for the boundaries 54.5 and 60.5 using the formula z = (x - μ) / σ. Then, find the probability corresponding to these z-scores using the standard normal distribution table or software.

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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 scenarios where outcomes are binary, such as success/failure or yes/no.

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. It involves adjusting the discrete values 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 sample size is small or the probability of success is not extreme.

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