Question

regression/correlation coefficient
the relationship between t, a students test scores, and d, the students success in college, is modeled by the equation d = 0.48t + 75.2. based on this linear regression model, the correlation coefficient could be

1. between - 1 and 0
2. between 0 and 1
3. equal to - 1
4. equal to 0

a linear regression equation of best fit between a students attendance and the degree of success in school is h = 0.5x + 68.5. the correlation coefficient, r, for these data would be

1. 0 < r < 1
2. - 1 < r < 0
3. r = 0
4. r = - 1

a nutritionist collected information about different brands of beef hot dogs. she made a table showing the number of calories and the amount of sodium in each hot dog.

calories per beef hot dog	milligrams of sodium per beef hot dog
181	477
176	425
149	322
184	482
190	587
158	370
139	322

a) write the correlation coefficient for the line of best fit. round your answer to the nearest hundredth.
b) explain what the correlation coefficient suggests in the context of this problem.

Explanation:

First - part:

The linear regression equation $d = 0.48t+75.2$ has a positive slope ($0.48$). A positive - slope in a linear regression equation implies a positive linear relationship between the variables. The correlation coefficient $r$ for a positive linear relationship lies between $0$ and $1$. So the answer for the first part is 2) between 0 and 1.

Second - part:

The linear regression equation $h = 0.5x + 68.5$ has a positive slope ($0.5$). A positive slope indicates a positive linear relationship. The correlation coefficient $r$ for a positive linear relationship is between $0$ and $1$, so the answer for the second part is 1) $0

Third - part (a):

Let $x$ be the calories per beef hot - dog and $y$ be the milligrams of sodium per beef hot - dog.

1. Calculate the means:

• $\bar{x}=\frac{186 + 181+176+149+184+190+158+139}{8}=\frac{1363}{8}=170.375$
• $\bar{y}=\frac{495 + 477+425+322+482+587+370+322}{8}=\frac{3480}{8}=435$

1. Calculate the standard deviations:

• $s_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$
• $\sum_{i=1}^{8}(x_i - \bar{x})^2=(186 - 170.375)^2+(181 - 170.375)^2+(176 - 170.375)^2+(149 - 170.375)^2+(184 - 170.375)^2+(190 - 170.375)^2+(158 - 170.375)^2+(139 - 170.375)^2$
• $s_x\approx18.17$
• $s_y=\sqrt{\frac{\sum_{i = 1}^{n}(y_i-\bar{y})^2}{n - 1}}$
• $\sum_{i=1}^{8}(y_i - \bar{y})^2=(495 - 435)^2+(477 - 435)^2+(425 - 435)^2+(322 - 435)^2+(482 - 435)^2+(587 - 435)^2+(370 - 435)^2+(322 - 435)^2$
• $s_y\approx96.97$

1. Calculate the covariance:

• $Cov(x,y)=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{n - 1}$
• $\sum_{i=1}^{8}(x_i - \bar{x})(y_i - \bar{y})=(186 - 170.375)(495 - 435)+(181 - 170.375)(477 - 435)+(176 - 170.375)(425 - 435)+(149 - 170.375)(322 - 435)+(184 - 170.375)(482 - 435)+(190 - 170.375)(587 - 435)+(158 - 170.375)(370 - 435)+(139 - 170.375)(322 - 435)$
• $Cov(x,y)\approx1653.88$

1. Calculate the correlation coefficient:

• $r=\frac{Cov(x,y)}{s_xs_y}=\frac{1653.88}{18.17\times96.97}\approx0.94$

Third - part (b):

The correlation coefficient $r\approx0.94$ is close to $1$. This suggests a strong positive linear relationship between the number of calories and the amount of sodium in beef hot - dogs. That is, as the number of calories in a beef hot - dog increases, the amount of sodium in the beef hot - dog also tends to increase.

Answer:

1. 2) between 0 and 1
2. 1) $0
3. (a) $r\approx0.94$

(b) There is a strong positive linear relationship between the number of calories and the amount of sodium in beef hot - dogs. As the number of calories increases, the amount of sodium tends to increase.