AI HL · 13.4
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Testing for correlation
13.4.17
Screenshot and video to come
Screenshot and video to come
A student measures the temperature of a lake at seven depths. Depth (m): 1, 3, 5, 8, 12, 15, 20; Temperature (°C): 19.3, 17.8, 20.1, 19.3, 17.7, 15.4, 13.9. Test, at the 5% significance level, whether there is negative correlation between depth and water temperature. State the assumption you make.
Have a go first. Then open the working one line at a time.
H0: ρ = 0, H1: ρ < 0
ρ is the population correlation coefficient between depth and water temperature. Negative correlation gives a lower-tail alternative.
From the calculator: linear regression t test with H1: ρ < 0: r = −0.861, p = 0.00636
Enter the two lists, choose the linear regression t test and set the alternative to ρ < 0.
0.00636 < 0.05
The p-value is below the significance level.
Reject H0. There is sufficient evidence at the 5% level of negative correlation between depth and temperature.
State it in the words of the question.
Depth and temperature are both normally distributed.
The p-value is only valid if both variables are normal.
A gardener grows nine identical plants under different amounts of light. Light (hours per day): 6, 8, 9, 10, 12, 13, 14, 16, 18; Growth (cm): 7.6, 9.0, 11.0, 11.1, 12.1, 14.2, 16.4, 17.7, 17.2. Find r, then test at the 1% significance level whether more light is linked to more growth.
Have a go first. Then open the working one line at a time.
H0: ρ = 0, H1: ρ > 0
ρ is the population correlation coefficient between hours of light and growth. More light, more growth: positive.
From the calculator: linear regression t test with H1: ρ > 0: r = 0.968, p = 0.00000906
The calculator gives r and p together.
0.00000906 < 0.01
The p-value is below 1%.
Reject H0. There is sufficient evidence at the 1% level of positive correlation between hours of light and growth.
Positive, as the question asked.
A food writer records the price and her rating of the lunch at eight cafés. Price (€): 6.5, 7.0, 8.5, 9.0, 10.5, 12.0, 13.5, 15.0; Rating (out of 10): 5.9, 6.6, 9.0, 8.0, 5.8, 8.0, 7.7, 8.9. Test, at the 5% significance level, whether there is any correlation between price and rating.
Have a go first. Then open the working one line at a time.
H0: ρ = 0, H1: ρ ≠ 0
"Any correlation" means either direction: two-tailed.
From the calculator: linear regression t test with H1: ρ ≠ 0: r = 0.500, p = 0.207
Choose the alternative ρ ≠ 0.
0.207 > 0.05
The p-value is above the significance level.
Do not reject H0. There is insufficient evidence at the 5% level of correlation between the price and the rating.
Say 'insufficient evidence', not 'there is no correlation'.
A teacher believes that the more time students spend on their phones, the lower their test mark. She records both for nine students. Phone use (hours per day): 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0, 6.0; Test mark (%): 67, 38, 63, 58, 67, 62, 71, 61, 59. Test her belief at the 5% significance level.
Have a go first. Then open the working one line at a time.
H0: ρ = 0, H1: ρ < 0
ρ is the population correlation coefficient between phone use and test mark. Her belief is negative correlation.
From the calculator: linear regression t test with H1: ρ < 0: r = 0.217, p = 0.713
The sample r is positive, so the data point away from H1 and the p-value is large.
0.713 > 0.05
The p-value is far above 5%.
Do not reject H0. There is no evidence at the 5% level of negative correlation between phone use and test mark.
The data do not support her belief at all.