相關系數：斯皮爾曼與皮爾遜：練習題

The least squares regression line for a dataset is given by the equation \(y = 15.2 + 0.8x\). What does the value \(0.8\) represent in the context of this model? <\/p>

A researcher is investigating the relationship between the age of a car (\(x\) years) and its current market value (\(y\), in thousands of pounds). For a sample of 12 cars, the following summary statistics were calculated:

\(\sum x = 60\)
\(\sum y = 144\)
\(\sum x^2 = 350\)
\(\sum y^2 = 1950\)
\(\sum xy = 620\)

Find the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + bx\).

A researcher calculates a least squares regression line \(y = a + bx\) and finds that the sum of the residuals, \(\sum(y_i - \hat{y}_i)\), for a sample of 20 points is exactly zero. They then identify one outlier with a very large positive residual. If this outlier is removed and the regression line is recalculated, what is the most likely effect on the new gradient \(b\) if the outlier had an \(x\)-value much higher than the mean \(\bar{x}\)?

For a specific data point in a regression analysis, the observed value is \(y = 25\) and the predicted value from the least squares regression line is \(\hat{y} = 22\). What is the value of the residual for this point? <\/p>

A researcher wants to test for a relationship between two variables. They decide to use Spearman's rank correlation coefficient instead of Pearson's product moment correlation coefficient (PMCC). Which of the following is a valid reason for this choice? <\/p>

In a linear regression model \(y = a + bx\), the sum of the squares of the residuals is minimized. Given a data point \((10, 25)\) and a regression line \(y = 5 + 1.8x\), calculate the residual for this point and interpret its meaning regarding the model's prediction.

A botanist is investigating the relationship between the amount of fertilizer used (\(x\), in grams) and the height of a specific species of plant (\(y\), in centimeters) after six weeks. The following data were collected from a sample of 10 plants:

\(\sum x = 120\)
\(\sum y = 185\)
\(\sum x^2 = 1650\)
\(\sum y^2 = 3540\)
\(\sum xy = 2380\)

(a) Calculate the Pearson product-moment correlation coefficient (\(r\)) for these data. (2 points)
(b) Determine the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + bx\). (2 points)
(c) A particular plant was given 15g of fertilizer and grew to a height of 22.5 cm. Calculate the residual for this plant and state whether the regression model overestimates or underestimates this plant's height. (2 points)

The daily sales of a small boutique (\(x\), in hundreds of pounds) and the daily temperature (\(y\), in \(^\circ C\)) were recorded for 12 days. The summary data is:
(a) Calculate the Pearson product-moment correlation coefficient. (2 points)
(b) A student suggests that the negative correlation proves that hot weather causes sales to drop. Critically evaluate this statement. (2 points)
(c) Calculate the gradient of the regression line of \(y\) on \(x\) and interpret it in the context of sales. (3 points)< /p>