z Scores. In Exercises 5–8, express all z scores with two deci...

Question

z Scores. In Exercises 5–8, express all z scores with two decimal places.

New York City Commute Time New York City commute times (minutes) are listed in Data Set 31 “Commute Times” in Appendix B. The 1000 times have a mean of 42.6 minutes and a standard deviation of 26.2 minutes. Consider the commute time of 95.0 minutes.

b. How many standard deviations is that [the difference found in part (a)]?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The recorded daily screen time usage in units of minutes for a group of individuals is analyzed. Among the 800 recorded times, the mean is 320.5 minutes with a standard deviation of 85.3 minutes. Given a screen time usage of 500 minutes. Determine what the Z score for the given screen time usage will be using the mean and standard deviation of the recorded times. Awesome. So it appears for this particular problem we're ultimately trying to figure out what the value for the Z score is, provided that we're told that we are given a screen time usage of 500 minutes and we're also told that we have a mean of 300. In 20.5 minutes and we're also told that we have a standard deviation of 85.3 minutes. And we're going to use all those data points to help us determine the Z score, and once again, the Z score is the final answer that we're ultimately trying to solve for. That's our end goal. So now that we know that we're, well, exactly we're trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is 2.45, B is 2.10. C is 1.95, and finally, D is 1.85. Awesome. So first off, in order to solve for this particular problem, let us take a moment to write down all of our known variables, all of our data points that are given to us or provided to us. So let us note that we're told that the recorded screen time, which we'll call this value X, X is equal to 500 minutes. We're also told the mean screen time, which will denote this as mu, and mu the mean screen time is equal to 320.5 minutes, and we're also told the standard deviation, which will denote this as sigma, and sigma is equal to 85.3. Minutes. Fantastic. So, our next step now. Our second step is we need to compute the difference. So let us know that the difference D is equal to x minus mu, which is going to be equal to me plug in or known variables, 500 minutes minus 320.5 minutes, which will mean that our difference D is equal to 179.5 minutes. And now our third and final step is we need to calculate how many standard deviations is the difference, or rather we need to calculate our Z score. So we need to recall and use our Z score equation or the equation to solve for the Z score, and the Z score Z is equal to x minus mu, our difference. Divided by our standard deviation sigma, which when we plug in our known data points, this is equal to 179.5 minutes. Divided by our standard deviation, which is 85.3 minutes, and as you can see, our units of minutes cancel out, leaving us with a unitless answer, which is perfect because all of our Z score values are without units. So that means our final answer when we plug it into our calculator and round the two decimal places like our multiple choice answers. We will see that it's going to be equal to 2.10, and that's it. That is our final answer. Hooray, we did it. And when we look at our multiple choice answers, this corresponds to multiple choice answer letter B, which once again is 2.10. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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Key Concepts

Z-Score

Mean

Standard Deviation

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