Sunny days, busy kiosk. How strong is the link?

A scatter diagram shows a pattern. One number measures how closely the points follow a straight line, and a test decides whether the pattern is more than chance.

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A kiosk records the hours of sunshine $x$ and the ice creams sold $y$ on eight days. The points rise from left to right, not perfectly, but clearly.

Cross the plot at the means, $\bar x = 5.5$ and $\bar y = 50.5$. A point above and right of centre, or below and left, has $(x - \bar x)(y - \bar y) > 0$: it pulls towards positive correlation. Six points do; the two in the other quadrants sit close to the centre and pull only a little.

The product moment correlation coefficient scales the sum of those products to lie between $-1$ and $1$. At $0.915$ it is close to $1$: strong positive linear correlation. A calculator gives it directly from the data.

Test $H_0: \rho = 0$ against $H_1: \rho > 0$ at the 1% level. For $n = 8$ the critical value is $0.7887$: the shaded region. $0.915$ lands inside it, so reject $H_0$. There is evidence of positive correlation between sunshine and sales.

A number for the pattern. A test for the population.

Two-tailed tests, p-values, what r can and can't tell you, straightening exponential and power curves with logarithms, and twelve graded exam questions with full worked solutions are waiting.

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