"Height versus Head Circumference Use the results of Problem 1...

Question

"Height versus Head Circumference Use the results of Problem 13 from Section 12.3 to answer the following questions:

c. Predict the head circumference of a randomly selected child who is 25.75 inches tall"

Accepted Answer

Hello. In this video, we are told that a small coffee shop tracked a daily average temperature and the number of hot coffees sold over 8 random days. Using the data above, we want to predict the number of hot coffee sold on a day when the average temperature is 12.5 °C. Now, in order to approach this problem, we're going to go ahead and use linear regression. The linear regression line is defined as the following. Y hat is equal to A plus B X. Here, B is going to be our slope, and A is going to be. A is going to be the white intercept. Now, the slope for the linear regression is defined as the following. B is equal to our sample size N multiplied by the sum of the product of X Y. Minus the sum of X multiplied by the sum of y. Divided by the sample size N multiplied by the sum of X2 minus the sum of X quantity squared. And our intercept A is defined as the following. A. It is equal To the mean of why. Minus B multiplied by the mean of X. So, the first thing we need to do is we need to go ahead and solve for our slope. So we need to solve for the following sums. We need the sums of the products of XY, we need the sum of X, the sum of Y, and we need the sum of X squares, and the sum of X quantity squared. So what we're going to do is we're going to order the data for the sums in an organized table. So what I provided here is the sums organized. So the first column represents the sum of Xs. So these are all the elements of X, and the total sum is going to be 108. The second column is the sums of Y's, and that is going to be a total sum of 848. The sum of X squares is 1500, and the sum of the products of XY is going to be 11,209. Here we have a total sample size of 8, so N is equal to 8, and by using these values, our slope B is going to be defined as 8 multiplied by 15, 1500, by 11,209. Minus the sum of X, which is 108, multiplied by the sum of Y, which is 848, and this will be divided by the sample size 8, multiplied by the sum of X squared, which is 1500 minus the sum of X, which is 108 quantity squared. Now, if we were to plug this into a calculator, we would get an approximate B value of 15.6905. So this is going to be the slope for our linear regression. Now, what we need to do is we need to solve for our constant value A. Now A requires the mean of Y and the mean of X. So let's go ahead and solve for the mean of X. Now, the mean of X is going to be the sum of all the data points divided by the sample size. So, the sum of all X data points is going to be 108 divided by 8, and this is going to give us a mean of 13.5. And for Y, that is going to be the sum of Y, which is 848 divided by 8, the sample size, and that is going to give us a total value of 106. So now that we have Y bar, and the average of Y and the average of X, and we have the value of B, we can solve for our constant. So A is equal to 13.5, to Y bar, which is 106 minus -5.6905, multiplied by 10, 13.5. And by plugging this into a calculator, we get an approximate constant value. Of A is equal to approximately 182.8218. So The linear regression line that we have now is going to be Y is equal. To 182.8218. Minus 5.6905 X. Now, remember, we are trying to predict the sales for when X is equal to 12.5 °C. So now we're gonna go ahead and plug in X is equal to 12.5. So doing this is going to give us 182.8218 minus 5.6905. Multiplied by 12.5, and by plugging this into a calculator, we get an approximate value of 111.69055. And rounding to the nearest whole number is going to give us the value of 112, which is going to be the solution to our problem. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

"Height versus Head Circumference Use the results of Problem 13 from Section 12.3 to answer the following questions:c. Predict the head circumference of a randomly selected child who is 25.75 inches tall"

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"Height versus Head Circumference Use the results of Problem 13 from Section 12.3 to answer the following questions:
c. Predict the head circumference of a randomly selected child who is 25.75 inches tall"