Use the same data as for the corresponding exercises in...

Use the same data as for the corresponding exercises in Section $10-1 .$ For each exercise, find the equation of the regression line and find the $y^{\prime}$ value for the specified $x$ value. Remember that no regression should be done when $r$ is not significant. Alumni Contributions Years and contribution data are as follows: $$ \begin{array}{l|cccccc} \text { Years } \boldsymbol{x} & 1 & 5 & 3 & 10 & 7 & 6 \\ \hline \text { Contribution } y, \$ & 500 & 100 & 300 & 50 & 75 & 80 \end{array} $$ Find $y^{\prime}$ when $x=4$ years.

Use the same data as for the corresponding exercises in Section $10-1 .$ For each exercise, find the equation of the regression line and find the $y^{\prime}$ value for the specified $x$ value. Remember that no regression should be done when $r$ is not significant.
Alumni Contributions Years and contribution data are as follows:
$$
\begin{array}{l|cccccc}
\text { Years } \boldsymbol{x} & 1 & 5 & 3 & 10 & 7 & 6 \\
\hline \text { Contribution } y, \$ & 500 & 100 & 300 & 50 & 75 & 80
\end{array}
$$
Find $y^{\prime}$ when $x=4$ years.

Key Concepts

Prediction using Regression

Prediction using regression involves substituting a specific value for the independent variable into the regression equation to obtain an estimated value of the dependent variable. This application is useful for forecasting outcomes based on observed trends in the data.

Regression Line Equation

The regression line equation, typically written as y = bx + a, expresses the predicted value of the dependent variable as a function of the independent variable. Here, 'b' represents the slope, indicating the change in the dependent variable for a one-unit change in the independent variable, and 'a' is the y-intercept, representing the predicted value when the independent variable is zero.

Significance Testing for Correlation

Significance testing for correlation involves determining whether the observed relationship between the independent and dependent variables is statistically significant and unlikely due to random chance. If the correlation is not statistically significant, it suggests that the linear relationship may not be reliable for prediction.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data. It provides a way to understand the strength and direction of the association between the variables.

Least Squares Method

The least squares method is a standard approach in regression analysis that minimizes the sum of the squared differences (residuals) between the observed values and the values predicted by the linear model. This optimization ensures that the regression line best fits the data.

Transcript

00:01 Here's a solution to the years and contributions data.

00:03 We're going to test first because we're supposed to find y hat, and then we're supposed to evaluate that whenever x equals 4.

00:08 But first we're going to test to make sure that r is significant.

00:11 So we have a null hypothesis that row equals 0, and then an alternative hypothesis that row does not equal 0, meaning we're assuming that it's not significant, but then we're trying to prove that it is.

00:21 And we need to look at r for that.

00:22 So if you go to stat, edit, there you can see your data values.

00:25 Okay, so if we go to stat, calc, and then lin, reg, tag, l1 is my x list, l2 is my y list.

00:32 And because i know what's going to happen, i'm going to go ahead and store this into y1.

00:37 All right, so i calculate, and that r is negative .883, which is a pretty good r.

00:41 All right.

00:42 So let's go ahead and write that down.

00:43 So we have r equals negative 0 .88.

00:46 I'll go ahead and go to four decimal places, 2 .9.

00:48 So let's find the test statistic, which is that r divided by the square root of 1 minus r squared.

00:55 So 1 minus .7795 divided by 1, 2, 3, 4, 6, so minus 2 equals 4.

01:05 And that's going to give us a test statistic of about negative 3 .76.

01:10 Okay, so now we need to find the p value here.

01:12 So the p value is if you go to stat, i'm sorry, a second of ours, and go to the tcdf, and we're just going to look at the left tail.

01:21 So i'm going to go from negative some large number, negative e99.

01:25 And in the upper boundary, i'm going to put my statistics.

01:28 So since it's negative, that's why i went to the left...

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