Finding Specified Data Values In Exercises 31–38, answer the q...

Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Weights of Teenagers In a survey of 18-year old males, the mean weight was 166.7 pounds with a standard deviation of 49.3 pounds. (Adapted from National Center for Health Statistics)

c. What weight represents the first quartile?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says in a survey of 17-year-old students, the mean number of hours spent on homework per week was 12.5 hours with a standard deviation of 4.2 hours. What is the first quartile or 25th percentile value for hours spent on homework? And we're given 4 possible choices as our answers. For choice A, we have 9.7 hours. For choice B, we have 8.7 hours. For choice C, we have 10.7 hours, and for choice D, we have 11.7 hours. Now, we're asked what is the first quartile or 25th percentile is a name for that value for hours spent on homework. Now, recall that for the 25th percentile, that our cumulative probability P is going to be equal to 0.25. And when we look up this cumulative probability in our tables, we see that it actually corresponds to a Z score with Z is equal to. Minus 0.674. So now we just need to use the Z score to figure out how many hours. And so recall your formula for the Z score, that is Z is equal to the quantity of X minus U in quantity divided by stigma. Where Z here is our Z score, X here is going to be the hours spent on homework, U is going to be the mean, and sigma is going to be the standard deviation. And we actually have values for the mean and standard deviation, so we'll substitute in those values. So for Z, we'll use the Z score that we found in our tables, which is gonna be minus 0.674, and that's going to be equal to the quantity of X, which is the test score that we're looking for. Minus our mu, which is our mean, we're told the problem of that is 12.5 hours, we'll subtract 12.5, and then we'll divide that quantity by our standard deviation sigma, which we're told the problem is 4.2 hours. And so now we just need to solve this equation for X. So the first step is to multiply both sides of the equation by 4.2. Doing so on the left hand side we'll have minus 2.8308, and that it is going to be equal to x minus 12.5. And so now all we need to do is to add 12.5 to both sides of the equation to solve for X. So doing so, we'll get that X is going to be equal to. 9.7. So here we just round it to one decimal place. And here the units for our value for X is going to be ours. Since both our mean and standard deviation head units of ours as well. So this here is the answer that we're looking for for this problem. And we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Normal Distribution

Quartiles

Standard Deviation

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