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

1:52 minutes

Problem 5.5.1

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=24, p=0.85, q=0.15

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 using the formula np = n × p. Substitute n = 24 and p = 0.85 into the formula to compute np.

Step 3: Calculate nq using the formula nq = n × q. Substitute n = 24 and q = 0.15 into the formula to compute nq.

Step 4: Check whether both np ≥ 5 and nq ≥ 5. If both conditions are satisfied, then the normal distribution can be used to approximate the binomial distribution.

Step 5: Conclude whether the normal approximation is valid 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=24 and p=0.85, 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 applicable 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 (24*0.85) and nq (24*0.15) meet this criterion.

Conditions for Normal Approximation

To use the normal distribution as an approximation for a binomial distribution, the sample size must be sufficiently large, ensuring that the distribution is not overly skewed. The rule of thumb is that both np and nq should be at least 5. This ensures that the binomial distribution is approximately symmetric and bell-shaped, making the normal approximation valid.

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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=24, p=0.85, q=0.15