Regression and Predictions
Exercises 13–28 use the same ...

Question

Regression and Predictions

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1.

Find the regression equation, letting the first variable be the predictor (x) variable.

Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5.

Cars Sales and the Super Bowl Listed below are the annual numbers of cars sold (thousands) and the numbers of points scored in the Super Bowl that same year. What is the best predicted number of Super Bowl points in a year with sales of 8423 thousand cars? How close is the predicted number to the actual result of 37 points?

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Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says, listed below are the annual number of pastry shops opened in hundreds and the number of goals scored in the World Cup that same year. What is the best predicted number of goals in a year with 84 pastry shops opened? How close is the predicted value to the actual results of 3 goals? So below the text we're given a table where the first column is denoted as year and it has values of 12345, and 6. The second column is labeled pastry shops in hundreds, and it has the values of 81, 82, 83, 90, 86, and 83. And the final column is goals, and it has the values of 24153, and 4. We're also given four possible choices as our answers. For choice A, we have 3.2 goals. The predicted value is very close to the actual value. For choice B, we have 2.8 goals. The predicted value is very close to the actual value. For choice C, we have 5.0 goals. The predicted value is very far from the actual value, and for choice D we have 7.2 goals. The predicted value is very far from the actual value. So, in order to answer this question, we need to find the equation for the least squares regression equation. So to do this, we'll need both the mean and standard deviation for our data for our pastry shops, as well as our goals. And so this is actually best done in a table. So one screen we have created a table, where Xabi is our first column, and this is the number of pastry shops in hundreds, and it has the values in this column of 81, 82, 83, 90, 86, and 83. And when we add all of those values up, we sum of all of our X values, we get 505. And for our mean, we'll take that 505 and divide it by our 6 years, and we get a value for our mean X bar of 84.167. Now for our next column, we have our Y values why sub I, and this corresponds to the goals scored. And here we have the values of 24153, and 4, and again summing our Y values. This is going to be equal to 19, and calculating our mean Y bar, by taking the 19 and divided by 6, we get 3.167. Now, in order to calculate our standard deviation, we'll need to first calculate the deviation from the mean for each of our data points. So, in the next column, we have the quantity of X I minus X bar, and when we calculate this for each of our points, we get the values of -3.167, minus 2.167, minus 1.167. 5.833, 1.833, and minus 1.167. In the next column we have Y I minus Y bar, and calculating this for each of our data points, we get the values of -1.167, 0.833, minus 2.167, 1.833, minus 0.167, and 0.833. The next column, we have the quantity of exabyte minus X bar in quantity squad. So we're just basically squaring the column of X by minus X bar, and we'll need this for our standard deviation. And so, in this column, we have the values of 10.03. 4.696, 1.362. 34.024, 3.36. And 1.362. And when we sum this column, we get a value of approximately 54.83. For the next column, we have the quantity of Y minus Y bar in quantity squared. And so this is just squaring our column of Y minus Y bar. In doing so, we get the values of 1.362, 0.694, 4.696, 3.360. 0.028. And 0.6. 94. And when we added that call, we get approximately 10.83. So now we have enough information to calculate our standard deviation. So, for x variable, our standard deviation S of X. is going to be equal to, and here we call your formula for the standard deviation, that's going to be the square root of the quantity of the sum of the quantity of X of I minus X bar in quantity squared, and that is divided by the quantity of N minus 1. And so this is going to be equal to the square root. Of the quantity of 54.83. Divided by the quantity of 6 minus 1. And evaluating this expression, this is going to be equal to 3.31. Now we're gonna do the same thing for our Y variable, so we'll have S of Y and that's going to be equal to the square root. Of the quantity of the sum. Of the quantity of yi minus y bar. In quantity squared, divided by the quantity of N minus 1. And so this is going to be equal to the square root of the quantity. Of 10.83. Divided by the quantity of 6 minus 1. And when we evaluate this expression, We get 1.47. And so, now we have all the information that we need in order to begin calculating the quantities that we're going to use for our regression equation. So the first thing we would calculate is our correlation coefficient, and we're gonna need one additional quantity for this. So, in the last column in our table, we have calculated the quantity of X by minus X bar in quantity, multiplied by the quantity of Y by minus Y bar in quantity. In doing this for each of our data points, we get the values of 3.696 minus 1.805, 2.529, 10.692. -0.306 and 0.972. And when we sum this column, we get approximately a value of 13.83, and we're actually going to need that for our correlation coefficient. So recall that your correlation coefficient are. is equal to The sum Of the quantity of X of I minus X bar in quantity multiplied by the quantity of YI minus Y bar in quantity. Divided by the square root of the quantity of the sum of the quantity of XOI minus X bar in quantity squared multiplied by the sum. Of the quantity of Y I minus Y bar in quantity squared in quantity. And so this is going to be equal to. For the numerator, we just calculated the value of 13.83, and that is going to be divided by The square root Of the quantity, and we had already calculated the two sums in the denominator underneath the square root sign. And so for the sum, For the X variables, we have 54. 083. And that gets multiplied by the sum of the quantity of Y minus Y bar in quantity square, and that will give us an 0.83. And so, evaluating this expression, this is going to be equal to. 0.568. And so now we can use this correlation coefficient. To calculate some of our 5th parameters. So, the first quantity that we'll calculate is going to be B1. Which is going to be the slope of our regression. So this is going to be equal to R multiplied by the quantity of S of Y divided by S of X. In quantity And so this is going to be equal to 0.568. Multiplied by the quantity And or as of why we had 1.47, and that's going to be divided by SFX which is 3.31 in quantity. And evaluating this expression, this is going to be equal to. 0.252. The next quantity that we need to calculate is going to be B knot, which is going to be our intercept, and so that's going to be equal to. Why bar? Minus B1 multiplied by X bar. So this is going to be equal to. For Y bar, we had 3.167, we'll have minus 0.252, multiplied by X bar, which is 84. Point 167. And evaluating this expression, this is going to be equal to. -18.0. So that means that our regression equation. Why hat is going to be equal to -18.0. Plus 0.252. Multiplied by X. And so now we're gonna use this equation. To predict the number of goals. And we're going to be using an x value of 84. So that means that white hat? is going to be equal to -18.0. Plus 0.252 multiplied by 84. And when we evaluate this expression, this is going to be equal to. 3.2. So that is going to be the answer for the first part of this question. And so now, if we compare our predicted number of goals of 3.2 to the actual number of goals, which is just 3.0, we see that the predicted value is 0.2 goals higher than the actual. So our predicted value is very close to the actual value. So that's gonna be the answer to the second half of this problem. So now if we come up here to our answer choices, we see that the correct answer choice actually corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Regression Analysis

Predictor and Response Variables

Prediction Procedure

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