Question

during the summer, rafaels toy shop collects data that implies a relationship between the weekly average temperature and the number of kites sold.
weekly temp °f (x) kites sold (y)
73.8 6
75.7 6
88.7 8
92.1 8
72.5 6
74.6 5

choose the correct equation for the least squares line.
a. $y=4.08x - 0.133$
b. $y=0.133x - 4.08$
c. $y=0.133x + 4.08$

during a week with an average temperature of 61.7 °f, how many kites can rafael expect to sell? (round to the nearest whole number as needed.)

during a week in which rafael sells 11 kites, what will be the approximate average temperature? (round to the nearest tenth as needed.)

the coefficient of correlation is . (round to the nearest thousandth as needed.)

Explanation:

Step1: Calculate mean of x and y

First, find $\bar{x} = \frac{73.8+75.7+88.7+92.1+72.5+74.6}{6} = \frac{477.4}{6} \approx 79.5667$
$\bar{y} = \frac{6+6+8+8+6+5}{6} = \frac{39}{6} = 6.5$

Step2: Compute numerator for r

Calculate $\sum (x_i-\bar{x})(y_i-\bar{y})$:
$(73.8-79.5667)(6-6.5) + (75.7-79.5667)(6-6.5) + (88.7-79.5667)(8-6.5) + (92.1-79.5667)(8-6.5) + (72.5-79.5667)(6-6.5) + (74.6-79.5667)(5-6.5)$
$= (-5.7667)(-0.5) + (-3.8667)(-0.5) + (9.1333)(1.5) + (12.5333)(1.5) + (-7.0667)(-0.5) + (-4.9667)(-1.5)$
$= 2.88335 + 1.93335 + 13.70 + 18.80 + 3.53335 + 7.45005 = 48.3$

Step3: Compute denominator for r

Calculate $\sum (x_i-\bar{x})^2$:
$(73.8-79.5667)^2 + (75.7-79.5667)^2 + (88.7-79.5667)^2 + (92.1-79.5667)^2 + (72.5-79.5667)^2 + (74.6-79.5667)^2$
$= 33.255 + 14.949 + 83.417 + 157.083 + 49.938 + 24.669 = 363.311$

Calculate $\sum (y_i-\bar{y})^2$:
$(6-6.5)^2 + (6-6.5)^2 + (8-6.5)^2 + (8-6.5)^2 + (6-6.5)^2 + (5-6.5)^2$
$= 0.25 + 0.25 + 2.25 + 2.25 + 0.25 + 2.25 = 7.5$

Denominator: $\sqrt{363.311 \times 7.5} = \sqrt{2724.8325} \approx 52.1999$

Step4: Calculate correlation coefficient

$r = \frac{48.3}{52.1999} \approx 0.925$

Answer:

0.925