Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 6.6.18a

Chapter 6, Problem 6.6.18a

Sleepwalking Assume that 29.2% of people have sleepwalked (based on “Prevalence and Comorbidity of Nocturnal Wandering in the U.S. Adult General Population, by Ohayon et al., Neurology, Vol. 78, No. 20). Assume that in a random sample of 1480 adults, 455 have sleepwalked.

a. Assuming that the rate of 29.2% is correct, find the probability that 455 or more of the 1480 adults have sleepwalked.

Verified step by step guidance

Step 1: Identify the type of probability distribution to use. Since we are dealing with a proportion (29.2%) and a large sample size (1480 adults), we can approximate the binomial distribution using the normal distribution. This is valid under the conditions that both np and n(1-p) are greater than 5, where n is the sample size and p is the probability of success.

Step 2: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = np, and the standard deviation is given by σ = √(np(1-p)). Substitute n = 1480 and p = 0.292 into these formulas.

Step 3: Convert the problem into a z-score calculation. To find the probability of 455 or more adults sleepwalking, we first calculate the z-score using the formula z = (X - μ) / σ, where X is the observed number of successes (455), μ is the mean, and σ is the standard deviation.

Step 4: Use the z-score to find the corresponding probability. Look up the z-score in a standard normal distribution table or use statistical software to find the cumulative probability up to the z-score. Since we are interested in 455 or more, subtract this cumulative probability from 1.

Step 5: Interpret the result. The final probability represents the likelihood of observing 455 or more adults sleepwalking in a random sample of 1480, assuming the true proportion of sleepwalkers is 29.2%.

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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. In this context, sleepwalking can be considered a 'success,' and the distribution helps calculate the probability of observing a certain number of successes (455) in a sample of 1480 adults, given the known probability of sleepwalking (29.2%).

Normal Approximation to the Binomial

For large sample sizes, the binomial distribution can be approximated by a normal distribution, which simplifies calculations. This approximation is valid when both np and n(1-p) are greater than 5. In this case, it allows us to use the normal distribution to find the probability of observing 455 or more sleepwalkers in the sample of 1480 adults.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, we can set up a null hypothesis that the true proportion of sleepwalkers is 29.2% and use the sample data to determine if the observed number of sleepwalkers (455) significantly deviates from what we would expect under this hypothesis, thus assessing the validity of the assumption.

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Step 1: Write Hypotheses

Related Practice

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