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For a random sample of 6 observations of pairs of values $(x, y)$, where $0 < x < 21$ and $0 < y < 14$, the following results are obtained - CIE - A-Level Further Maths - Question 10 - 2016 - Paper 1
Question 10
For a random sample of 6 observations of pairs of values $(x, y)$, where $0 < x < 21$ and $0 < y < 14$, the following results are obtained. \[ \sum x^2 = 844.20, \qu... show full transcript
Worked Solution & Example Answer:For a random sample of 6 observations of pairs of values $(x, y)$, where $0 < x < 21$ and $0 < y < 14$, the following results are obtained - CIE - A-Level Further Maths - Question 10 - 2016 - Paper 1
Step 1
Find the product moment correlation coefficient for the sample.
Answer
To find the product moment correlation coefficient (), we first calculate the necessary sums:
[ n = 6, \quad \sum x = 12.10, \quad \sum y = 50.40 ]
Now, we can use the formula for the correlation coefficient:
[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{\left( n \sum x^2 - (\sum x)^2 \right) \left( n \sum y^2 - (\sum y)^2 \right)}} ]
Substituting the values in:
[ r = \frac{6 \cdot 625.59 - 12.10 \cdot 50.40}{\sqrt{\left(6 \cdot 844.20 - (12.10)^2\right) \left(6 \cdot 481.50 - (50.40)^2\right)}} ]
Calculating gives:
[ r \approx 0.986 ]
Step 2
Find the equations of the regression lines of y on x and x on y.
Answer
To find the regression line of on , we calculate the slope (): [ b_{yx} = \frac{r \cdot S_y}{S_x} ] Where and are the standard deviations calculated from variances. Given that: [ S_x = \sqrt{36.66} \quad \text{and} \quad S_y = \sqrt{9.69} ] The slope calculates to: [ b_{yx} \approx 0.507 ] Thus, the regression equation is: [ y = a + b_{yx}x ] Finding : [ a = \bar{y} - b_{yx} \bar{x} ] For the regression line of on similarly: [ b_{xy} = \frac{r \cdot S_x}{S_y} \ ] Thus, the regression equation is: [ x = a' + b_{xy}y ]
Step 3
Use the appropriate regression line to estimate the value of x when y = 6.4 and comment on the reliability of your estimate.
Answer
From the regression line on , after finding and substituting :
[ x = \frac{y - a}{b_{yx}} ]
Substituting gives:
[ x \approx 6.36 ]
Considering reliability, we check the correlation coefficient:
Given , the estimate is reliable as it is close to 1, indicating a strong linear relationship.