In a least squares regression, what is the name for the difference between the observed value and the fitted value?

A nutritionist investigates whether there is a linear association between daily protein intake (grams) and muscle strength score in a sample of $12$ adults. From the sample, she computed the sample correlation coefficient and obtained a $p$-value of $0.012$ for testing the population correlation. State the conclusion for the hypotheses at the $5\%$ significance level.

The following data represents the number of hours of exercise per week ($x$) and the corresponding cholesterol levels ($y$) of individuals. Use this data to determine the slope of the least-squares regression line.

A researcher developed a regression model to predict the final exam score $\left(y\right)$ based on the number of hours studied $\left(x\right)$. The regression equation is $\hat{y}=50+2x$. The sample size is $n=16$, the mean number of hours studied is $\bar{x}=10$, $S_{xx}=240$, and the standard error of estimate is $s_y=5$. Construct a $90\%$ prediction interval for the final exam score when a student studies for $14$ hours.

A researcher collects data on students' favorite colors and their preferred study locations. Is it reasonable to use a least-squares regression line to analyze this data?

A nutritionist predicts daily calorie intake ($y$) based on hours of physical activity per week (x) using the regression equation $\hat{y}=1800-50x$. The standard error of estimate is $s_y=150$, the sample size is $n=25$, the mean hours of activity is $\bar{x}=5$, and $S_{xx}=250$. Construct a $90\%$ prediction interval for a person who exercises $x_0=8$ hours per week.

A researcher is studying the relationship between the number of hours employees spend in professional training sessions per month ($x$) and their job performance ratings ($y$), scored out of $100$. A linear regression model is created to predict job performance based on training hours, and the resulting regression equation is $\hat{y}=2.8x+65$. The researcher collects data from a sample of $10$ employees. Based on this data:

Residual sum of squares (SSR): $\sum (y-\hat{y}{)}^2=82.4$

Total sum of squares (SST): $\sum (y-\bar{y}{)}^2=240.0$

What is the coefficient of determination $r^2$, and what does it tell us about the model?

Which statement best describes the meaning of a point $(x,\hat{y})$ on the least-squares regression line?

Which statement correctly describes the number of dependent and independent variables in a multiple regression equation?

A regression model for predicting annual revenue from number of employees is based on data where x ranges from 20 to 100. Is it appropriate to use the model to predict revenue for x = 150?