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Inferential Statistics · Paper 3 — Free Maths AA HL Practice Questions

1ChallengeLAQCorrelation and regression analysis8 marksPaper 3~12 min

A researcher investigates the relationship between hours of sunlight per day,

x

(hours), and crop yield,

y

(tonnes per hectare), for a sample of

n = 8

farms. The data are summarised as:

\sum x_i = 48, \quad \sum y_i = 56, \quad \sum x_i^2 = 320, \quad \sum y_i^2 = 412, \quad \sum x_i y_i = 360

(a)

Show that the Pearson product-moment correlation coefficient for this data is

r = \dfrac{3}{\sqrt{10}}

. [4 marks]

(b)

(i) The researcher uses the critical value

r_{\text{crit}} = 0.707

for a two-tailed test at the

5\%

significance level with

n = 8

. State the hypotheses and determine whether the correlation is statistically significant at this level. [2]
(ii) An alternative approach uses the test statistic

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

which follows a

t

-distribution with

n - 2

degrees of freedom under

H_0

. Calculate

t

and hence verify the conclusion from (b)(i). Evaluate whether the researcher's claim of statistical significance is reliable, with reference to the assumptions required for Pearson's correlation coefficient to be valid. [4 marks]

Solutions

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