In Exercises 1–4, the sample size n, probability of success p, an...

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 5.5.3

Textbook Question

In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.
n=18, p=0.90, q=0.10

Verified step by step guidance

Step 1: Recall the rule for using a normal distribution to approximate a binomial distribution. The approximation is valid if both np ≥ 5 and nq ≥ 5, where n is the sample size, p is the probability of success, and q is the probability of failure.

Step 2: Calculate np by multiplying the sample size n by the probability of success p. Use the formula: np = n × p.

Step 3: Calculate nq by multiplying the sample size n by the probability of failure q. Use the formula: nq = n × q.

Step 4: Check whether both conditions np ≥ 5 and nq ≥ 5 are satisfied. If both conditions are met, the normal approximation can be used; otherwise, it cannot.

Step 5: Conclude whether the normal distribution can approximate the binomial distribution based on the results of the calculations in Steps 2 and 3.

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Key Concepts

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

Binomial Distribution

A 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). In this case, with n=18 and p=0.90, we can analyze the likelihood of achieving a certain number of successes in these trials.

Normal Approximation

The normal approximation to the binomial distribution is used when certain conditions are met, specifically when both np and nq are greater than or equal to 5. This allows us to use the normal distribution to estimate probabilities for binomial experiments, simplifying calculations. In this scenario, we need to check if np and nq meet this criterion to determine if the normal approximation is valid.

Conditions for Normal Approximation

To use the normal approximation for a binomial distribution, the conditions np ≥ 5 and nq ≥ 5 must be satisfied. Here, np = 18 * 0.90 = 16.2 and nq = 18 * 0.10 = 1.8. Since nq is less than 5, the normal approximation is not appropriate for this binomial experiment, indicating that we should rely on the binomial distribution for accurate probability calculations.

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In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.n=18, p=0.90, q=0.10