A retail company's year-end share price (in units of dollars) is modeled by $y=-60+6.85x_1-1.92x_2$, where $x_1$ is total assets (in units of billions of dollars) and $x_2$ is liabilities (in units of billions of dollars). Predict the share price when $x_1=22.5$ and $x_2=10.4$.

A researcher studying the relationship between hours spent practicing piano and performance score on a music exam found the regression line $\hat{y}=8.25x+62.5$. Predict the performance score for a student who practiced for $3$ hours.

What is the main goal of the least squares method in linear regression?

A researcher uses the regression equation $\hat{y}=5+0.6x$ to predict the number of hours studied ($y$) based on the number of classes attended ($x$). The standard error of the estimate is $s_e=2.5$, the mean number of classes attended is $\bar{x}=15$, $n=12$, and $\sum (x_i-\bar{x}{)}^2=120$. Construct a $99\%$ prediction interval for a student who attended $20$ classes.

A behavioral scientist examines whether there is a linear relationship between weekly meditation minutes and perceived stress score in a sample of $15$ participants. The test of the population correlation produced a $p$-value of $0.032$. State the conclusion for the hypotheses at the $5\%$ significance level.

Which of the following residual patterns suggests that the regression model is NOT valid?

In a study examining the relationship between study hours and test scores, a linear regression line was fitted to the data. One student studied for $6$ hours and scored $82$, while the average test score of all students was $75$. Which of the following best describes the total variation for this student?

The table shows the average vehicle age (in years) for each of the eight years. A regression model has been fit to the data:

$\hat{y}=0.0893x-168.45$

Using this regression model and the actual data, compute the standard error of estimate ${𝑠}_{𝑒}$, which measures the typical distance between observed and predicted values.

If a residual plot displays a clear curved pattern, what does this indicate about the suitability of a linear regression model?

Suppose the linear correlation coefficient for a dataset is $r=0.72$. What is the coefficient of determination, $R^2$? Interpret the result.

Suppose the correlation coefficient for a set of data is $r=0.910$. What is the coefficient of determination? How much of the variation is explained and how much is unexplained?

Why is R squared always non-negative, even if r is negative?

A regression model for predicting monthly sales from number of employees yields R squared = 0.05. What does this suggest?

Using the regression equation $\hat{y}=5.2+2.3x$ to predict the number of visitors at a museum based on advertising dollars spent (in units of thousands), find the $90\%$ prediction interval for $x=30$ if $s_y=4.5$, $n=12$, $\bar{x}=28$, and $\sum (x_i-\bar{x}{)}^2=48$. Use the appropriate $t^{\ast }$ value for $10$ degrees of freedom.

A random sample of $130$ male engineering students is obtained, and their heights are measured in $\text{inches}$ and in $\text{centimeters}$.

From the regression analysis, the predicted height in $\text{centimeters}$ for a student who is $70\text{ inches}$ tall is found to be $177.8\text{ cm}$. The sum of squared residuals from the regression is ${\text{42.25 cm}}^2$.

i. What is the value of the standard error of estimate (SE)?

ii. For a student whose height is $70\text{ inches}$, what is the $95\%$ prediction interval?