Example Average life expectancy and processor speed of the CPU in PCs varied over:

Year	Processor speed (Hz)	Life expectancy (years)
1990	$6.6 \times 10^7$	74
1995	$1.2 \times 10^8$	75
2000	$1.0 \times 10^9$	77
2005	$1.8 \times 10^9$	79
2010	$2.4 \times 10^9$	80

Test at the 5% significance level whether there is positive correlation between processor speed and life expectancy.

This means we are trying to show with 95% certainty (or 5% uncertainty) whether correlation exists.

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1. Define the parameter $\rho$ ("rho") in words

Let $\rho$ = "the population correlation coefficient between processor speed and life expectancy of a PC."

(Must be in context).

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2. State the null and alternate hypotheses

$H_0: \rho = 0 \quad \text{(Assume no correlation)}$

$H_1: \rho > 0 \quad \text{(Testing for positive correlation)}$

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3. Calculate 'r' from the test data

$\Rightarrow r = :highlight[0.9896]$

The table gives us "critical values" for different significance levels.

If testing is in a given direction (in this case we are only testing for positive correlation), the tables tell us the cut-off point for no correlation. Anything bigger than this value (or smaller than the negative of the value if testing for negative correlation), then we can say with '$x$%' certainty (where $x = 100$ - sig level) that there is this type of correlation.

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4. Compare your observed value for '$r$' to the critical value in the table

$$0.9896 > 0.8054$$

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5. Conclude in context

Reject $H_0$.

Sufficient evidence to suggest that there is a positive correlation between processor speed and life expectancy of a PC.

Example A road safety group test the braking distance of cars of different ages

Age in years	Braking distance in metres
3	31.3
6	38.6
7	40.1
7	35.1
9	48.4

Test at the 10% significance level whether there is a correlation between age and braking distance.

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Let $\rho$ be the population correlation coefficient between the age of a car and its braking distance.

$H_0: \rho = 0$

$H_1: \rho \neq 0$

We test $\rho \neq 0$ as we have not been told to test for + or - correlation.

Note: Because we have no idea in which direction to test, we split the 10% significance: 5% testing for positive and 5% for negative correlation.

$$r = :highlight[0.8756]$$$$0.8756 > 0.8054$$

Reject $H_0$.

Sufficient evidence to suggest that there is a correlation between age and braking distance of a car.

This means that $r > 0.8054$ or $r < -0.8054$ would lead to rejection of $H_0$.

Example: Find Spearman's rank correlation coefficient for each of the following data sets.

Conduct a two-tailed test for association using a 10% significance level.

x	46	32	22	53	73	31	37
y	75	38	37	70	26	92	26

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Note that Spearman's rank tests use the method except:

• There is a different table to read from.
• The hypotheses refer to "association" rather than $\rho$.