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 25 24 y 60 95 140 174 180 r critical r Conclusion Reject the null hypothesis, there is sufficient evidence to show that age and weight of Shetland ponies are correlated. Reject the null hypothesis, there is insufficient evidence to show that age and weight of Shetland ponies are correlated. Fail to reject the null hypothesis, there is insufficient evidence to show that age and weight of Shetland ponies are correlated. Fail to reject the null hypothesis, there is sufficient evidence to show that age and weight of Shetland ponies are correlated. (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 975 992 935 976 936 y 40 100 65 145 74 149 r critical r Conclusion Reject the null hypothesis, there is sufficient evidence to show that lowest barometric pressure and maximum wind speed for cyclones are correlated. Reject the null hypothesis, there is insufficient evidence to show that lowest barometric pressure and maximum wind speed for cyclones are correlated. Fail to reject the null hypothesis, there is insufficient evidence to show that lowest barometric pressure and maximum wind speed for cyclones are correlated. Fail to reject the null hypothesis, there is sufficient evidence to show that lowest barometric pressure and maximum wind speed for cyclones are correlated.

Answer 1

a).

We first calculate the correlation for the given data by forming the following table:

	X	Y	x^2	y^2	x*y
	3	60	9	3600	180
	6	95	36	9025	570
	12	140	144	19600	1680
	25	174	625	30276	4350
	24	180	576	32400	4320
Total	70	649	1390	94901	11100

The correaltion between X and Y is

The value of r=0.9633

The critical value of r for is 0.878. Since r>the critical value, we reject the null Hpothesis>

r =0.9633

critical r =0.878

Conclusion

Reject the null hypothesis, there is sufficient evidence to show that age and weight of Shetland ponies are correlated.

b). The given data with squares and cross products is:

	X	Y	x^2	y^2	x*y
	1004	40	1008016	1600	40160
	975	100	950625	10000	97500
	992	65	984064	4225	64480
	935	145	874225	21025	135575
	976	74	952576	5476	72224
	936	149	876096	22201	139464
Total	5818	573	5645602	64527	549403

r =-0.9826

critical r =0.917

Conclusion

Reject the null hypothesis, there is sufficient evidence to show that lowest barometric pressure and maximum wind speed for cyclones are correlated.