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

1:51 minutes

Problem 5.5.8

Textbook Question

In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.
P(x<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. Continuity correction is used when approximating a discrete distribution (like binomial) with a continuous distribution (like normal).

Step 2: Recall the rule for continuity correction. When converting a discrete value to a continuous range, adjust the value by ±0.5 depending on the inequality. For example, P(x < k) in a binomial distribution becomes P(x < k - 0.5) in the normal distribution.

Step 3: Apply the continuity correction to the given binomial probability statement P(x < 109). Since the inequality is 'less than', subtract 0.5 from 109. This gives P(x < 108.5) in the normal distribution.

Step 4: Match the corrected normal distribution probability statement P(x < 108.5) with the corresponding option provided in the problem. From the options, this matches option (b).

Step 5: Verify the process by reviewing the continuity correction rule and ensuring the adjustment aligns with the inequality direction ('less than' in this case). This confirms the correct match is P(x < 108.5).

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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 to improve accuracy.

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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.P(x<109)a. P(x>109.5)b. P(x<108.5)c. P(x<109.5)d. P(x>108.5)