Questions: Question 6, 4.3.17 The following data represent the number of calories per serving and the number of g of high-fiber cereals. Complete parts (a) through (d) below. Calories, x 220 240 150 190 180 Sugar, y 16 15 17 17 13 (a) A scatter diagram of the data is shown. What type of relation appears to exist between calories and sugar content? A. There appears to be little or no relationship between calories and sugar content. B. There appears to be a strong linear relationship between calories and sugar content. C. There appears to be a strong nonlinear relationship between calories and sugar content. (b) Determine the correlation coefficient between calories and sugar content. The correlation coefficient is -0.204.

Solution Steps

To solve the given problem, we need to follow these steps:

(a) Analyze the scatter diagram to determine the type of relationship between calories and sugar content.

(b) Calculate the correlation coefficient between calories and sugar content using the given data.

(c) Determine the critical value for the correlation coefficient from the provided table of critical values.

Given the scatter diagram, we need to determine the type of relationship between calories and sugar content. The options are:

• A. There appears to be little or no relationship between calories and sugar content.
• B. There appears to be a strong linear relationship between calories and sugar content.
• C. There appears to be a strong nonlinear relationship between calories and sugar content.

The correlation coefficient \( r \) between calories and sugar content is given as: \[ r = -0.204 \]

To determine the critical value for the correlation coefficient, we use the following parameters:

• Number of data points, \( n = 5 \)
• Significance level, \( \alpha = 0.05 \)
• Degrees of freedom, \( \text{df} = n - 2 = 3 \)

Using the t-distribution table, the critical value \( t_{\text{critical}} \) for \( \alpha = 0.05 \) and \( \text{df} = 3 \) is: \[ t_{\text{critical}} = 3.1824 \]

The critical value for the correlation coefficient \( r_{\text{critical}} \) is calculated as: \[ r_{\text{critical}} = \sqrt{\frac{t_{\text{critical}}^2}{t_{\text{critical}}^2 + \text{df}}} = \sqrt{\frac{3.1824^2}{3.1824^2 + 3}} = 0.8783 \]

Final Answer