Question 890067: Recently, I have been taking an online Finite Math course. I have been having problems with this chapter and especially the question below. Can someone please help?
At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed of a particular player was 104 miles per hour (mph)and the standard deviation of the serve speeds was 8 mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was mound-shaped and symmetric. What proportion of the player's serves was between 112 mph and 120 mph?
Answer by Okamiden(22)   (Show Source):
You can put this solution on YOUR website! (This is going to be a lenghty explanation, sorry if you wanted something more straight to the point.)
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First, you need a Normal values table. It's tricky, because there are two ways your table can be arranged. The answer is the same, but the interpretation is slightly different...
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In these kinds of problem, the solution is always found the same way. First, you find out how far your values are from the mean.
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Here, you have a mean of 104, and your two values are 112 and 120. 112 is 8 points away from 104 and similarly, 120 is 16 points away.
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Next, to be able to use a Normal table, you will want to know how many standard deviations away your values are from the mean.
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Here, 112 is 8 points away from the mean 104, and ONE standard deviation is exactly 8 points! That means 112 is exactly one standard deviation away from your mean. What about 120? It's 16 points away. 16/8=2, which means our 16 points is exactly worth two standard deviations. More specifically, the values we just found (1 and 2) are called the Z-values, or Z-scores, and they're all we're going to need from now on. You can forget everything else, just remember that we're looking for how many people fall between Z=1 and Z=2!
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Now here comes the tricky part, using a normal table. Here's a link to the table I'm using for your problem:
http://www.sjsu.edu/faculty/gerstman/EpiInfo/z-table.htm
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How to interpret the table: The column on the right is simply your Z-score, up to 1 decimal precision. The row on top provide added precision. For example, if we had gotten a Z-score of 1.96, we'd locate 1.9 on the column and 0.06 on the top row.
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What we have, though, is Z=1.00 and Z=2.00. Z=1.00, in the table, is 0.8413. Z=2.00 is 0.9772.
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What do these values represent? The 0.8413 means that 84.13% of people are below 1 standard deviation. In real words, 84.13% of people serve slower than 112mph. They could serve at 0mph, or 110, or 104, but 84.13% are below 112mph.
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Similarly, 0.9772 means that 97.72% of people are below Z=2, which means they serve below 120mph.
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If we know 97.72% of people serve below 120mph, and we know that 84.13% of people serve below 112mph, how many people serve at EXACTLY between 112mph and 120mph?
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That would be 97.72%-84.13%=13.59%. More precisely, 13.59% of these tennis players achieved an average serve speed between 112mph and 120mph.
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I know this is super confusing, and it seems hard, but with practice it becomes more natural. Good luck!
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