Questions: For the accompanying data set, (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient. and (c) determine whether there is a linear relation between x and y. (a) Draw a scatter diagram of the data. Choose the correct graph below. (b) By hand, compute the correlation coefficient: The correlation coefficient is r= (Round to three decimal places as needed.) (c) Determine whether there is a linear relation between x and y. Because the correlation coefficient is and the absolute value of the correlation coefficient, ]. is than the critical value for this data set, linear relation exists between x and y.

Solution Steps

The provided scatter plot in the problem corresponds to the data set. It shows the relationship between x and y values. The graph increases from left to right, suggesting a positive correlation.

The data set consists of the following points: (2,6), (4,10), (6,12), (6,14), (8,4).

Here's how to calculate the correlation coefficient (r):

1. Calculate the mean of x and y:

   • Mean of x (x̄) = (2+4+6+6+8)/5 = 5.2
   • Mean of y (ȳ) = (6+10+12+14+4)/5 = 9.2

2. Calculate the deviations from the mean for x and y:

   • For x: (-3.2, -1.2, 0.8, 0.8, 2.8)
   • For y: (-3.2, 0.8, 2.8, 4.8, -5.2)

3. Calculate the product of the deviations for each pair:

   • (10.24, -0.96, 2.24, 3.84, -14.56)

4. Sum the products of deviations:

   • ∑[(x-x̄)(y-ȳ)] = 10.24 - 0.96 + 2.24 + 3.84 - 14.56 = 0.8

5. Calculate the sum of squares of deviations for x and y:

   • ∑(x-x̄)² = 10.24 + 1.44 + 0.64 + 0.64 + 7.84 = 20.8
   • ∑(y-ȳ)² = 10.24 + 0.64 + 7.84 + 23.04 + 27.04 = 68.8

6. Calculate the correlation coefficient (r):

   • r = ∑[(x-x̄)(y-ȳ)] / √[∑(x-x̄)² * ∑(y-ȳ)²]
   • r = 0.8 / √(20.8 * 68.8)
   • r = 0.8 / √1430.4 ≈ 0.8 / 37.82 ≈ 0.021

The calculated correlation coefficient is approximately 0.021. The critical value for n=5 (from a critical values table) at a 0.05 significance level (commonly used) is approximately 0.878. Since the absolute value of the calculated correlation coefficient (0.021) is less than the critical value (0.878), there is no statistically significant linear relationship between x and y.

Final Answer