Comparing z-Scores from Different Data Sets The table shows po...

Question

Comparing z-Scores from Different Data Sets The table shows population statistics for the ages of Best Actor and Best Supporting Actor winners at the Academy Awards from 1929 to 2020. The distributions of the ages are approximately bell-shaped. In Exercises 51–54, compare the z-scores for the actors.

Best Actor 1970: John Wayne, Age: 62
Best Supporting Actor 1970: Gig Young, Age: 56

tab

Best Actor 1970: John Wayne, Age: 62

Best Supporting Actor 1970: Gig Young, Age: 56

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 table below shows the population statistics for ages of Olympic gold medalists in the 100 m sprint and the marathon from 1950 to 2020. The distributions are approximately normal. 100 m sprint is in our left column and our right column is the marathon. And in our 100 m sprint column, we have mu is approximately equal to 25.4 years, and sigma is approximately equal to 3.1 years. In our marathon column, we have mu is approximately equal to 30.8 years, and sigma is equal to 4.7 years. In 2016, the 100 m sprint gold medalist was 34 years old, and the marathon gold medalist was 28 years old, who had a more unusual age for their event. Awesome. So it appears for this particular problem we're trying to determine ultimately who had a more unusual age for their event. Was it the 100 m sprint gold medalist who was 34 years old, or was it the marathon gold medalist who was 28 years old? So with that in mind, our first step once again is we need to note that we're trying to determine which gold medalists had a more unusual age for their event, and in order to do that, we will need to recall and use the Z score or how to use a Z score. And apply that to each athlete. And as we should recall, the Z score tells us how many standard deviations an individual's age is from the mean age for that event for this particular problem. And as we should recall, the formula to solve for the Z score is Z, the Z score is equal to x minus mu divided by sigma, where X is the age of the individual, mu is the mean age, and sigma is the standard deviation. So now we need to calculate the Z score for both athletes. So, I have to keep it simple, I'll just call the 100 m sprint gold medalist just 100. So for the 100 m sprint, we're told that X is equal to 34. We're also told that the mean mule is approximately equal to 25.4, and we're also told that sigma is approximately equal to 3.1. Awesome. So, to keep things simple, I'm leaving off the units, but note that for our ages, they're all in units of years. Awesome. So, let me plug in these values into our equation, you will find that Z. Our Z score is approximately equal to 34 minus 25.4, all divided by 3.1, which will give us an answer of approximately when we round to two decimal places, 2.77. Fantastic. So now, focusing our attention on our second answer that we're trying to solve for for the marathon runner, which I'll just call M for short. And we're told that X for this case for the marathon runner, is equal to 28, because they're 28 years old. We're also told that the mean mu is equal to 30.8, and we're also told that sigma is approximately equal to, so it's important to note, I put equals, but it's not, we need to note that mu is approximately 30.8 and sigma is approximately 4.7. So once again, plugging in our known variables to solve for our Z score, we will find that Z is approximately equal to 28 minus 30.8 divided by 4.7. Which will be approximately equal to when we round to two decimal places, -0.60, fantastic. So, at this point, we need to recall and note that a higher absolute value of the Z score will indicate a more unusual or less typical age, and we need to note that the absolute value of 2.77 is greater than the absolute value of -0.60. So that will mean therefore, as a result, the 100 m sprint gold medalist with a Z score of approximately 2.77 had a much more unusual age. Compared to the marathon gold medalist whose Z score is about -0.60, so that will mean that our final answer, without a doubt, will have to be once again. The 100 m sprint gold medalist had a more unusual age for their event, and that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Z-Score

Mean (μ)

Standard Deviation (σ)

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