Seven Days of Iced Drinks

A café owner logs one week of data: each day's highest temperature and the number of iced drinks sold. Across the $n = 7$ days, Pearson's correlation is $r = 0.71$. That looks like a strong positive relationship, and the owner is ready to plan stock around it.

With so few days, though, chance alone can produce a large $r$. To check, test

• $H_0$: there is no linear relationship (the true correlation is 0), against
• $H_1$: there is one, in either direction (a two-sided test),

at significance level $\alpha = 0.05$. The test statistic is

$$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$

and when $H_0$ is true it follows a $t$ distribution with $df = n - 2$ degrees of freedom. $H_0$ is rejected when $|t|$ is at least the critical value $t^*$:

With 7 days of data, what is the smallest value of $|r|$ that this test would declare significant?

Round your answer to 3 decimal places.