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 21 24 y 60 95 140 190 172 r Incorrect: Your answer is incorrect. critical r Incorrect: Your answer is incorrect. 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. Correct: Your answer is correct. (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 975 992 935 970 924 y 40 100 65 145 65 146 r critical r Incorrect: Your answer is incorrect. 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)

Following table shows the calcualtions:

	X	Y	X^2	Y^2	XY
	3	60	9	3600	180
	6	95	36	9025	570
	12	140	144	19600	1680
	26	190	676	36100	4940
	19	180	361	32400	3420
Total	66	665	1226	100725	10790

Sample size: n = 5

The coefficient of correlation is :

Critical value of r for n=5 is 0.88.

Since r > 0.88 so relationship between the variables is significant.

Correct option:

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

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(b)

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	1004	40	1008016	1600	40160
	975	100	950625	10000	97500
	992	65	984064	4225	64480
	935	145	874225	21025	135575
	970	65	940900	4225	63050
	924	146	853776	21316	134904
Total	5800	561	5611606	62391	535669

Sample size: n = 6

The coefficient of correlation is :

Critical value of r for n=6 is 0.92

Since |r| > 0.92 so relationship between the variables is significant.

Correct option:

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