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A radar unit is used to measure the speed of automobiles on an expressway during rush-hour traffic. The speeds of individual automobiles are normally distributed with a mean of $62 \mathrm{mph}$.

a. Find the standard deviation of all speeds if $3 \%$ of the automobiles travel faster than $72 \mathrm{mph}$.

b. Using the standard deviation found in part a, find the percentage of these cars that are traveling less than $55 \mathrm{mph}.$

c. Using the standard deviation found in part a, find the 95 th percentile for the variable "speed."

a. $5.32 \mathrm{mph}$

b. $9.34 \%$

c. $70.75 \mathrm{mph}$

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{'transcript': "Okay, so we need to confuse you a few things here. The first thing that's us to find is the mean. This is absolutely a weighted mean. That's because there's a certain frequency by which the cars are going at 47 or 52 miles an hour and you'll notice that 40 seven's not in the book. I went ahead and put the middle points of each of those little ranges. So this was I believe, I believe, I believe from 45 to 49 this is from 50 to 54. And so I went ahead and find the middle points for all those because that is what we're going to use and feel free to look above. It gives all the formulas that you're supposed to stick in middle point and multiply it by. This frequency is essentially to find this mean and so I like to go sheets do all that for me. I quick average, dot waited, selected this comma and then selected this calm and in common. And then this column and it gave me that average of 60.68 miles per hour, which sounds right about in the middle there, so seems to make sense. Next, it took quite a bit of time to find standard deviations were here. Follow with me. Well, you need to do is you needed multiple because again, this is weighted right there. Certain, like 10 cars going about 47. Where is 175 are going about 62 miles an hour. So here's how you do it. And again, this formula is, I believe, about two pages prior. And it is, I believe, called sample variance. So feel free to check. Anyway, You take this frequency and you multiply that times this middle point subtracted by me. Then you square that whole thing. And so that's exactly what I did here and the quick way to do this. I clicked. Enter and there's a little boxer. If I drag that box down, it will do it for all seven of these Rose and I got all those values. Next you need to add all the months that's meeting some of all this. And then finally, you need to take that number and divide it by the number of occurrences here. And so that's what I did right here. Divided by the number of occurrences there, and we get a variance of about 31.16 And of course, the standard deviation is just the square root of that. And so I typed in S Q R t to get the square to that number, and there it is, the variance in standard deviation with these different kind of waited sort of things going on there."}

University of Denver

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