Question

Mr. Bookman, a new fiction author, wishes to know whether there is any relationship between a book’s length and the price charged for the book. He chooses a random sample of 10 novels and records the length and price of each as follows:

Length (pages) 520 680 740 200 400 800 750 500 300 350
Price (Ksh 00) 16 20 21 9 10 22 20 15 10 10

a) Is there a significant relationship between novel length and its price? (use test for significance of correlation)

b) Develop a regression model for predicting the price of a novel using its length.

c) Test the significance of the model using t-test, R2 and F-test.

Answer 1

a) the correlation coefficient is calculate using correlation function in Excel

$$CORREL (array1, array2) = 0.977$$

The following needs to be tested:

$$Null: \rho = 0$$

$$Alternative:\rho\ne0$$

The sample size = 10. This means DF = n-2 =10 -2 =8

The corresponding critical correlation value $r_c$ for a significance level of  $\alpha =0.05$, for a two-tailed test is:

$r_c =0.632$

Observe that in this case, the null hypothesis $H_0: \rho = 0$ is rejected if $|r| >r_c =0.632$

we have that $|r| =0.977 > r_c=0.632$ , from which is concluded that the null hypothesis is rejected. There is a significant correlation.

b) Regression from Excel toolPak has following output predicting prices

The regression model equation: $Price (Ksh) =2.424 length (pages) + 259.729$

c) Testing the model

$R^2$ indicates there is goodness-of-fit because 95.44% of the variation on prices is contributed by the model. Overall, the model is statistically is significant since $F(1,8) =167.08, p-value < 0.05$ . The regression coefficient for length is also statistically significant because the corresponding t-stat has a $p-value < 0.05.$