In Exercises 5–8, match the binomial probability statement with i...

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

1:47 minutes

Problem 5.5.6

Textbook Question

In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.
��(��≥109)

a. P(x>109.5)
b. P(x<108.5)
c. P(x<109.5)
d. P(x>108.5)

Verified step by step guidance

Step 1: Understand the problem. The task is to match a binomial probability statement with its corresponding normal distribution probability statement after applying a continuity correction. The binomial statement given is P(x ≥ 109).

Step 2: Recall the concept of continuity correction. In a binomial distribution, x is discrete, but when approximating it with a normal distribution, which is continuous, we adjust the boundaries by ±0.5 to account for the discrete-to-continuous transition.

Step 3: Apply the continuity correction to the given binomial statement P(x ≥ 109). To include the value 109 in the normal distribution, we adjust the boundary to P(x > 108.5). This ensures that the probability includes all values starting from 109 and above.

Step 4: Match the corrected statement P(x > 108.5) with the options provided. From the list of options, the correct match is (d) P(x > 108.5).

Step 5: Conclude that the binomial probability statement P(x ≥ 109) corresponds to the normal distribution probability statement P(x > 108.5) after applying the continuity correction.

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Key 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 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). Understanding this distribution is crucial for analyzing scenarios where outcomes are binary, such as success/failure or yes/no.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution is used when the number of trials is large, allowing the binomial probabilities to be approximated by a normal distribution. This is particularly useful because normal distributions are easier to work with mathematically. The approximation is valid when both np and n(1-p) are greater than 5, ensuring that the distribution is not too skewed.

Continuity Correction

Continuity correction is applied when using a normal distribution to approximate a discrete distribution, such as the binomial. It involves adjusting the discrete values by 0.5 to account for the fact that the normal distribution is continuous. For example, to find P(X ≥ k) in a binomial distribution, one would use P(X > k - 0.5) in the normal approximation.

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In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.��(��≥109)a. P(x>109.5)b. P(x<108.5)c. P(x<109.5)d. P(x>108.5)