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Problem 44 Hard Difficulty

The following table gives the number of days in June and July of recent years in which the temperature reached 90 degrees or higher in New York’s Central Park. Source: The New York Times and Accuweather.com.
a. Prepare a frequency distribution with a column for intervals and frequencies. Use six intervals, starting with 0–4.
b. Sketch a histogram and a frequency polygon, using the intervals in part a.
c. Find the mean for the original data.
d. Find the mean using the grouped data from part a.
e. Explain why your answers to parts c and d are different.
f. Find the median and the mode for the original data.

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Video Transcript

this question chooses a table. Um, that describes how maney days reached 90 degrees in New York's Central Park. Um, first, it wants us to make a frequency table group thes by bins of length five, starting with 04 So we'll have 045 to 9 10 to 14 and want six of them. So we'll go up to 15 to 1922 24 and finally 25 to 29. Now, the way I'm gonna do is I'm gonna go through and for each one, I'll just make a little mark. Um, by the been, that number falls in. So we start 11 8 11 38 So to do that, I will make a mark where 11 goes here 10 to 14 that mark where it goes 11 8 11 than three and then eight. And I'll keep doing that. I've actually already done it. And I'll find that the frequencies we have our nine in the first been nine in the second, 14 15 and one. So these are our frequencies. Um, now wants me to make a hist a gram and a okay, frequency polygon. So I'm gonna drop my axes here. Um, and the way I'm gonna set up my bins is I'm gonna have them left endpoint. Inclusive. So this is gonna be zero. And if this is for the bar that's here, will include zero, but not Excuse me. I'll have this to be five. So the bar here will include zero, but not five. And that's what we want, cause the next barbell include five. So if we do that, I will have my been set up like this. 05 10 15 20 25 and 30. Um, and I'll choose to scale my ex excess up. Teoh. Let's just go by twos. Up to seven. So, too for six. Heat 10 12 14. Time in 14 not seven. All right. And so now we can start drawing this out. So in our first been or 04 we head nine. So that will be 2468 and up to nine right there in our second Been. We also had nines. That'll be right here in our third. We had 14 is all the way to the top at 14 there, and our fourth really had one, then spike up to five and then back down to one. From here, we can draw our frequency. Polygon. This is just connecting the mid points of the tops of each bar. So it'll look something like this is hand drawn. There's not gonna be perfect, but we'll have a curve that looks like this will give us a give us a shape. Um, that sort of described the shape of how, uh, these numbers are distributed. Now, the next thing he wants us to do is find, um the mean I was just find the mean What we do to find the mean we know is we sum up all of our data all of our exes, and defied by the number When I was a calculator to count everything up, I'm gonna get that. We have 300 80. Uh, that the sum is 380. Um, and we know that we have 39 data points just from counting. So dividing one by the other will get that the mean. The exact meat is nine point 74 Now, the problem wants us to use the frequency table that we just made to find the mean We know that frequent finding the meat from frequency table we have export is equal to the sum of X times the frequency over some of the frequencies or end. We ready? Know that And is 39 doesn't really matter. But we need what is gonna be X Here we have our values, our ranges. So when our values air ranges, that means we need to take the mid points. Um, so our midpoint, they're gonna be as follows. We have midpoint to then seven, then 12 17 and so on up until 27 and all right, down our frequencies again. So these are exes. Our frequencies are 99 14 15 and one. Now we can make a column. Another column is gonna be X times F and X times f is gonna look like this. 18 63 168 17 10. And you're not 10 110 and 27. So now that we have X times, African adult is up and I have done this in the calculator. Another x times F Some of these is gonna equal 403. So x bar is going to be 403 divided by 39 which is equal to uh, let's see. That's gonna be good to 10.33 Now what happened here? How is this different? We see we have an exportable tonight, boys. And Ford, an expert with a 10.33 next question, wants us to know how. How are these reconcilable? How can we have two different average is Well, what happens is this average found from the frequency table is just an estimate. It's an estimate because we took a bunch of values in our been. So we had several value nine values in our 0 to 4 range. They're all over and we said, Let's just pretend that they're all too So we changed the data a little bit by making them all uniform. And that's how our answered changes just a little bit. Now, finally, this question wants us to answer which the median and what is the mode? Will the median we know we have an equal to 39 and so when we have an odd number of data, points are median is the point at 39 plus 1/2, which is 40 over to which is 20 our 20th data point sorted from least the highest or highest released. Um, and then we count that out. We'll find that our median is actually 10. And finally the mode, the mode, we know it's not very obvious. But when recounting it out, we see that we have five instances of 11 and ah, no other value shows of that many times, so our mode is equal to 11.