The following are values of the product moment correlation coefficient between the x and y values of three different large samples of bivariate data. State what each indicates about the appearance of a scatter diagram illustrating the data. (i) -1. (ii) 0.02. (iii) 0.92. (b) In 1852 Dr William Farr published data on deaths due to cholera during an outbreak of the disease in London. The table shows the altitude (in feet, above the level of the river Thames) at which people lived and the corresponding number of deaths from cholera per 10000 people. Altitude, x 10 20 50 70 90 100 350 Number of deaths, y 102 63 22 27 21 10 0 ${\Sigma x = 700, \Sigma y = 149000, \Sigma x^2 = 275, \Sigma y^2 = 17351, \Sigma xy = 134040.}$ (i) Calculate the product moment correlation coefficient. (ii) Test, at the 5% significance level, whether there is evidence of negative correlation.

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The following are values of the product moment correlation coefficient between the x and y values of three different large samples of bivariate data - CIE - A-Level Further Maths - Question 9 - 2010 - Paper 1

Question 9

The following are values of the product moment correlation coefficient between the x and y values of three different large samples of bivariate data. State what each... show full transcript

Worked Solution & Example Answer:The following are values of the product moment correlation coefficient between the x and y values of three different large samples of bivariate data - CIE - A-Level Further Maths - Question 9 - 2010 - Paper 1

Step 1

State what each indicates about the appearance of a scatter diagram illustrating the data.

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Answer

For each of the correlation coefficients:

(i) -1. This indicates a strong negative correlation. In a scatter diagram, this would be represented as a straight line that decreases from left to right, indicating an inverse relationship between x and y.

(ii) 0.02. This value shows an almost negligible linear relationship. The scatter diagram would be quite scattered with no clear trend, suggesting that x and y are nearly independent of each other.

(iii) 0.92. This value indicates a strong positive correlation. The scatter diagram would show a straight line with a steep incline, reflecting that as x increases, y also significantly increases.

Step 2

Calculate the product moment correlation coefficient.

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Answer

To calculate the correlation coefficient, we apply the formula:

$r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n(\Sigma x^2) - (\Sigma x)^2][n(\Sigma y^2) - (\Sigma y)^2]}}$

Here, we have:

n = 7 (number of observations)
${\Sigma x} = 700$
${\Sigma y} = 149000$
${\Sigma x^2} = 275$
${\Sigma y^2} = 17351$
${\Sigma xy} = 134040$

Thus, plugging these values into the equation:

$r = \frac{7(134040) - (700)(149000)}{\sqrt{[7(275) - (700)^2][7(17351) - (149000)^2]}}$

Calculating this gives:

$r = -0.636$

Step 3

Test, at the 5% significance level, whether there is evidence of negative correlation.

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Answer

The hypothesis test can be set up as:

Null Hypothesis (H0): ( \rho = 0 ) (no correlation)
Alternative Hypothesis (H1): ( \rho < 0 ) (negative correlation)

We compare the calculated r value to the critical value from the Pearson correlation coefficient table.

Using a significance level of 0.05, and degrees of freedom (df = n - 2 = 5), the critical value is approximately -0.669.

If the calculated correlation r is less than -0.669, we reject the null hypothesis.

Since our calculated r = -0.636 is greater than -0.669, we do not reject the null hypothesis. There is no sufficient evidence to conclude that there is negative correlation.

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The following are values of the product moment correlation coefficient between the x and y values of three different large samples of bivariate data - CIE - A-Level Further Maths - Question 9 - 2010 - Paper 1

Worked Solution & Example Answer:The following are values of the product moment correlation coefficient between the x and y values of three different large samples of bivariate data - CIE - A-Level Further Maths - Question 9 - 2010 - Paper 1

State what each indicates about the appearance of a scatter diagram illustrating the data.

Calculate the product moment correlation coefficient.

Test, at the 5% significance level, whether there is evidence of negative correlation.

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