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For a random sample of ten pairs of values of x and y taken from a bivariate distribution, the equations of the regression lines of y on x and of x on y are, respectively - CIE - A-Level Further Maths - Question 8 - 2015 - Paper 1

Question 8

For a random sample of ten pairs of values of x and y taken from a bivariate distribution, the equations of the regression lines of y on x and of x on y are, respect... show full transcript

Worked Solution & Example Answer:For a random sample of ten pairs of values of x and y taken from a bivariate distribution, the equations of the regression lines of y on x and of x on y are, respectively - CIE - A-Level Further Maths - Question 8 - 2015 - Paper 1

Step 1

(i) Find the value of the product moment correlation coefficient for this sample.

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Answer

To find the product moment correlation coefficient (r), we use the formula:

$r = \sqrt{b_{xy} \times b_{yx}}$

From the regression equations, we extract the slopes:

For y on x, the slope (b_xy) is 0.38.
For x on y, the slope (b_yx) is 0.96.

Now, substituting the values:

$r = \sqrt{0.38 \times 0.96} = \sqrt{0.3648} \approx 0.604$

Thus, the value of the product moment correlation coefficient is approximately 0.604.

Step 2

(ii) Using a 5% significance level, test whether there is positive correlation between the variables.

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Answer

To test for positive correlation, we set up the following hypotheses:

Null Hypothesis (H0): (\rho = 0) (no correlation)
Alternative Hypothesis (H1): (\rho > 0) (positive correlation)

Using a significance level of 5%, we need to find the critical value of r from the correlation significance table for n-2 degrees of freedom where n is 10 (thus 8 degrees of freedom). The critical value for a one-tailed test at the 5% level is approximately 0.549.

Since our calculated r (0.604) is greater than the critical value (0.549), we reject the null hypothesis, indicating that there is significant evidence of positive correlation between the variables.

Step 3

(b) Find the least possible value of n.

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Answer

In this case, we are given that the product moment correlation coefficient is 0.507. To find the least possible value of n for significant correlation:

Using the formula for the critical value:

$r_{0.05, n-2} = 0.497$

From this, we set up the inequalities:

If n = 16: (r_{0.05, 14} = 0.497)
If n = 15: (r_{0.05, 13} = 0.514)

Since the correlation coefficient of 0.507 is greater than 0.497, the least possible sample size n satisfying the significance level is 16.

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Questions answered

For a random sample of ten pairs of values of x and y taken from a bivariate distribution, the equations of the regression lines of y on x and of x on y are, respectively - CIE - A-Level Further Maths - Question 8 - 2015 - Paper 1

Worked Solution & Example Answer:For a random sample of ten pairs of values of x and y taken from a bivariate distribution, the equations of the regression lines of y on x and of x on y are, respectively - CIE - A-Level Further Maths - Question 8 - 2015 - Paper 1

(i) Find the value of the product moment correlation coefficient for this sample.

(ii) Using a 5% significance level, test whether there is positive correlation between the variables.

(b) Find the least possible value of n.

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