Chapter 4.1, Problem 21P

Expand Your Knowledge: Using a Table to Test $\rho$ The correlation coefficient r is a sample statistic. What does it tell us about the value of the population correlation coefficient $\rho$(Greek letter rho)? We will build the formal structure of hypothesis tests of $\rho$ in Section 11.4. However, there is a quick way to determine if the sample evidence based on r 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 $\rho \ne 0$. We do this by comparing the value |r| to an entry in Table 4-6. The value of $\alpha$ in the table gives us the probability of concluding that $\rho$ =0 when, in fact, $\rho$= 0 and there is no population correlation. We have two choices for a: $\alpha :\alpha =0.05\text{ or }\alpha =\mathrm{0.01.}$ TABLE 4-6 Critical Values for Correlation Coefficient r

n	$\alpha = 0.05$	$\alpha = 0.01$	n	$\alpha = 0.05$	$\alpha = 0.01$	n	$\alpha = 0.05$	$\alpha = 0.01$
3	1.00	1.00	13	0.53	0.68	23	0.41	0.53
4	0.95	0.99	14	0.53	0.66	24	0.40	0.52
5	0.88	0.96	15	0.51	0.64	25	0.40	0.51
6	0.81	0.92	16	0.50	0.61	26	0.39	0.50
7	0.75	0.87	17	0.48	0.61	27	0.38	0.49
8	0.71	0.83	18	0.47	0.59	28	0.37	0.48
9	0.67	0.80	19	0.46	0.58	29	0.37	0.47
10	0.63	0.76	20	0.44	0.56	30	0.36	0.46
11	0.60	0.73	21	0.43	0.55
12	0.58	0.71	22	0.42	0.54

(a) Look at Problem 13 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 $\alpha =0.05$.

(b) Look at Problem 15 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 $\alpha =0.01$.