Sleepwalking Assume that 29.2% of people have sleepwalked (bas...

Question

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.

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

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says research indicates that 23.8% of adults have a pet cat. And they ran a sample of 1250 adults, 350 reported owning a cat. Assuming the true portion is 23.8%, what is the probability that 350 or more adults in the sample own a cat? And we're given four possible choices as our answers. For choice A, we have 0.99997. For choice B, we have 0.4895. For choice C, we have 0.0003, and for choice D, we have 0.2380. So, the first thing we need to do is recognize that we have a binomial distribution, since we only have two choices, either the person can own a cat, or they do not own a cat. So, we're going to write down a couple of properties for our binomial distribution. First thing we're going to do is write down the probability of owning a cat. So, we'll label that as P, and this is going to be equal to, and we're given that in a problem that it's going to be the 23.8%. We'll actually need this as a decibel, so we'll divide that by 100, and we'll get the 0.238. We also need the number of points in our sample, which is going to be in, and we're also given that in the problem, that is the 1250. Next, we need to calculate our mean, which is mute. And recall that this is going to be equal to N multiplied by P. Which is going to be equal to the 1250, multiplied by 0.238. Which is going to be equal to. 297.5. Next, we need to calculate our standard deviation, sigma, which is going to be equal to, if you recall, the square root of the quantity of N multiplied by P, multiplied by the quantity of 1 minus P in quantity. So this is going to be equal to the square root of the quantity of 1250, multiplied by 0.238. Multiplied by the quantity of 1 minus 0.238 in quantity. And when we evaluate this expression, this is going to be equal to. 15.056. So now we're going to do is use the normal approximation with continuity correction. That means we'll be calculating a Z score. So recall that Z is equal to quantity of X minus U in quantity divided by sigma, where X here is the point that we're looking for. So here we're wanting the probability. That 350 Or more adults have a cat. So we're actually going to use for our X value, 349.5. So that means our C score is going to be equal to 349.5 minus our average. Which are mean, uh, which is 297.5, and that quantity is divided by our standard deviation, which is 15.056. And when we evaluate this expression, this is going to be equal to. 3.45. So now what we need to do is use our Z tables to figure out the probability that Z is going to be greater than 3.45, but our Z tables are typically for Z less than a value. So if we look at our Z table, we see that the probability P of Z less than or equal to 3.45. Is going to be equal to. 0.9997. So, to get the probability that Z is greater than 3.45, which would give us the probability that 350 or more adults in our sample own a cat. Then to get that probability, we'll have P of Z greater than 3.45. And that's gonna be equal to 1 minus P of Z less than or equal to 3.45. Which is going to be equal to 1 minus 0.9997, which is equal to 0.0003. So this here is the answer that we're looking for for this problem. And if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Binomial Distribution

Normal Approximation to the Binomial

Hypothesis Testing

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