Finding Sample Statistics In Exercises 15 and 16, find the ran...

Question

Finding Sample Statistics In Exercises 15 and 16, find the range, mean, variance, and standard deviation of the sample data set.

Pregnancy Durations The durations (in days) of pregnancies for a random sample of pregnant people

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267 281 286 269 264 299 275

Accepted Answer

Welcome back, everyone. The following are the weights in grams of apples picked from a tree. 152, 160, 148, 155, 162, 150, 158, 149, 161, 154, 156, and 153. Calculate the range, mean, variance and standard deviation for this sample data set. Now let's start with our range. Recall that the range is the difference between the maximum and the minimum values for our data set, and in this data set, notice that the minimum value is 148 while the maximum value is 162. So that means the range equals 162 minus 148, which is 14. Therefore, 14 g is the range of the data set, the data set, or 14 g is the range of the weights of apples picked from this tree in our sample data set. Next, let's calculate the sample mean. Now the sample mean, OK, the sample mean here. Which is gonna be X bar. Is equal to the sum of values divided by the number of values. In this case, notice that we have 12 values. So we're gonna add all of them up, 152, 160, 148, 155, 162, 150. 158, 149, 161, 154, 156, and 153, OK. So we'll add all of these up and then we're gonna divide them by the number of values which is 12. When we add all of those values, it equals 1,858 and 1858 divided by 12 is gonna be approximately equal to 154.8. Thus the mean is approximately 154.8 g. Next, let's move on to the variants. What do we know about the sample variants? Well, recall that the sample variants. A sample variance. Which is gonna be a squared. Is gonna be equal to the sum of the squared deviations, so that's the sum of X minus X bar squared, all divided by N minus 1, OK? So, no, notice that if we're gonna find the sample variances, we'll need to find the sum of the squared deviations. So, let's create a table to help us find the deviations and then square those deviations so that we can add them up, OK? So a table makes it a bit easier to read. So here we have each value X. We have the difference between our value and our sample mean, and then we're going to square that difference. X minus X bar or squared. And no, that means in our table we can put all of our values from our data set, OK? Let me just put this here and let me just bring this over here a little bit, OK. Just to make some space. And I remember that in our data set our values are 152, 160, 148, 155, 162, 150. 1:58. 149, 161. 154. Um, 156 and 153, OK? So we can, we can pretty much stop here. So now what are we gonna do about our values? Oh, let me carry this down a little bit. What do we know? Well, starting with 152, the square, first, this deviation actually is going to be 152, OK, minus. Our sample mean, which we know is 1,858 divided by 12. I know that equals -176 or 2.833. And when we square that, it's going to be 8.028. Not to find the rest of the values, essentially, we're going to do the same thing. And I'll leave you in the interest of time to figure out what those deviations are, what for what, or to calculate the deviations. But for 160, you should get the deviation to be 5.167, and thus it's squared to be 26.694. For 148, it's gonna be negative 6.8333 and 4, sorry. 6.833 and 46.694. For 155, it's gonna be 0.167 and 0.028. 162, uh, that's 7.167, and 51.361, 150, that's negative 4.833, and 23.361. 158 is 3.167. And 10.028, 149, that's negative 5.833, and 34.028. 150. Sorry, I'm at 1, I'm at 161. OK, that's gonna be 6.167. And 34.028, 154, it's negative 0.833 and 0.694. 156 is 1.167. And 1.361, and 153 is negative 1.833, and 3.361. OK? So now we have all of our square deviations, and to find the sum, all we need to do is to add these up. Now when we do that, we should get the sum of the squared deviations to be approximately equal to 243.667. So now if we go back to our formula for the sample variants, that means, OK, if I continue it here, it's going to be equal to. 243.667 divided by 12 minus 1 we know 243.667 is an approximate value. So when we add it here, or sorry, when we divide it by 12 minus 1, we should get approximately 22.2, OK? So this is the sample variance. Finally, let's try to figure out our sample standard deviation. Now, the standard deviation is really just uh the square root of the variance, OK? In other words, a standard deviation S equals the square root of Squared. So if we find a square root here of 243.667 divided by 12 minus 1, which is 11, then we should get it to be approximately 4.7. So what does this tell us in summary? Now we know that our range is 14, our sample mean is 154.8, the sample variance is 22.2, and the standard deviation is 4.7. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Descriptive Statistics

Mean

Variance and Standard Deviation

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