Question

A researcher working for the Ray-Ban Corporation decides to perform a hypothesis test to investigate whether there is a significant positive correlation
between monthly temperature and number of sunglasses sold at one of their locations.  
The researcher randomly selected the following sample from this location.  
Monthly Temperature (°F)                    Number of Sunglasses Sold
                        80                                                   91
                     61                                                   34
                     56                                                   66
                     99                                                   72
                     71                                                   74
                   95                                                  125
                     43                                                    65

Using the coefficient of determination, which statement is correct?

	a.	37% of the variability in the number of sunglasses sold per month can be accounted for by the variability in the monthly temperature.
	b.	57% of the variability in the monthly temperature can be accounted for by the variability in number of sunglasses sold per month.
	c.	72% of the variability in the monthly temperature can be accounted for by the variability in number of sunglasses sold per month.
	d.	61% of the variability in the number of sunglasses sold per month can be accounted for by the variability in the monthly temperature.
	e.	45% of the variability in the monthly temperature can be accounted for by the variability in number of sunglasses sold per month.

Answer 1

Correlation coefficient calculation

Step 1: Find XY, X and Y2 as it was done in the table below. X 80 Y 91 34 66 61 56 99 71 X Y 7280 2074 3696 7128 5254 11875 2795 X - X 6400 3721 3136 9801 5041 9025 1849 Υ.Υ 8281 1156 4356 5184 5476 15625 4225 72 74 125 95 43 65 Step 2: Find the sum of every column to get: ΣΧ = 505 , ΣΥ= 527, ΣΧ.Υ = 40102, ΣΧ? = 38973, ΣΥ? = 44303 Step 3: Use the following formula to work out the correlation coefficient. η- ΣΧΥ – ΣΧ.ΣΥ » ΣΧ- (Σκ] - [ΣΥ? - (ΣΥ] 7 - 40102 – 505 - 527 T 0.6074 [7- 38973 – 5052] : [7 - 44303 – 527?]

r = 0.6074

Coefficient of Determination = 0.6074*0.6074 = 0.37

Thus, correct answer is as follows:

a. 37% of the variability in the number of sunglasses sold per month can be accounted for by the variability in the monthly temperature.