Question

The correlation coefficient r is a sample statistic. What does it tell us about the value of the population correlation coefficient ρ (Greek letter rho)? You do not know how to build the formal structure of hypothesis tests of ρ yet. However, there is a quick way to determine if the sample evidence based on ρ is strong enough to conclude that there is some population correlation between the variables. In other words, we can use the value of r to determine if ρ ≠ 0. We do this by comparing the value |r| to an entry in the correlation table. The value of α in the table gives us the probability of concluding that ρ ≠ 0 when, in fact, ρ = 0 and there is no population correlation. We have two choices for α: α = 0.05 or α = 0.01.

(a) Look at the data below regarding the variables x = age of a Shetland pony and y = weight of that pony. Is the value of |r| large enough to conclude that weight and age of Shetland ponies are correlated? Use α = 0.05. (Use 3 decimal places.)

x	3	6	12	16	23
y	60	95	140	188	182

r
critical r

(b) Look at the data below regarding the variables x = lowest barometric pressure as a cyclone approaches and y = maximum wind speed of the cyclone. Is the value of |r| large enough to conclude that lowest barometric pressure and wind speed of a cyclone are correlated? Use α = 0.01. (Use 3 decimal places.)

x	1004	975	992	935	974	928
y	40	100	65	145	66	151

r
critical r

Answer 1

a)

S.No	X	Y	(x-x̅)2	(y-y̅)2	(x-x̅)(y-y̅)
1	3	60	81.0000	5329.00	657.0000
2	6	95	36.0000	1444.00	228.0000
3	12	140	0.0000	49.00	0.0000
4	16	188	16.0000	3025.00	220.0000
5	23	182	121.0000	2401.00	539.0000
Total	60	665	254.0000	12248.00	1644.0000
Mean	12.000	133.00	SSX	SSY	SXY

correlation coefficient r=

Sxy/(√Sxx*Syy) =

0.932

critical r = 0.878 (please try 0.88 if this comes wrong)

b)

S.No	X	Y	(x-x̅)2	(y-y̅)2	(x-x̅)(y-y̅)
1	1004	40	1296.0000	2970.25	-1962.0000
2	975	100	49.0000	30.25	38.5000
3	992	65	576.0000	870.25	-708.0000
4	935	145	1089.0000	2550.25	-1666.5000
5	974	66	36.0000	812.25	-171.0000
6	928	151	1600.0000	3192.25	-2260.0000
Total	5808	567	4646.0000	10425.50	-6729.0000
Mean	968.000	94.50	SSX	SSY	SXY

correlation coefficient r=

Sxy/(√Sxx*Syy) =

-0.967

critical r = 0.917 (please try 0.92 if this comes wrong)