A Physics teacher takes a random sample of size 5 from his students - CIE - A-Level Further Maths - Question 10 - 2016 - Paper 1

To test for positive correlation, we state the null and alternative hypotheses:

• Null Hypothesis (H0): $\rho \leq 0$
• Alternative Hypothesis (H1): $\rho > 0$

Next, we will calculate the critical value of r from the correlation table. Using a one-tailed test at the 5% significance level with 5 samples:

• The critical value for r is approximately 0.878.

Since our calculated value is $0.796$, which is less than the critical value of $0.878$, we fail to reject the null hypothesis, indicating no positive correlation.

For the combined sample, we calculate:

• $\Sigma x = 96$, $\Sigma y = 86$
• $\Sigma x^2 = 1096$, $\Sigma y^2 = 1220$

Using these, we find: $\bar{x} = \frac{96}{12} = 8$ $\bar{y} = \frac{86}{12} \approx 7.17$

Now calculate the gradient (p) and intercept (q) for the regression line: $p = \frac{S_{xy}}{S_{xx}} \text{ and } q = \bar{y} - p\bar{x}$.

Substituting the known values will yield:

• $y = px + q$.

Using the derived regression equation to estimate the practical test score for x = 13:

• Substitute x = 13 into the regression line equation obtained previously.

Consider the reliability of the estimate by analyzing the correlation coefficient (r) computed earlier. Since $r = 0.796$, indicating a moderate correlation, we can state that the estimate is reliable.

• Thus, the estimated score for the practical test will fall within the range of calculated predictions, making the estimate reasonably reliable.