A sample of 50 students records their weekly study hours (X) and exam scores (Y). The sample mean of X is 12 hours with variance $s_{x}^{2}=9$. The mean of Y is 78 with variance $s_{y}^{2}=64$. The sum of products $\sum (X_{i}-\bar X)(Y_{i}-\bar Y)$ equals 180.

(a) Compute the Pearson correlation coefficient $r$.

(b) Determine the least‑squares regression line predicting Y from X in the form $\hat Y = a + bX$.

(c) At the 5 % significance level, test whether the slope $b$ differs from zero (use a two‑tailed t‑test with $n-2$ degrees of freedom). State the conclusion.

Question

A sample of 50 students records their weekly study hours (X) and exam scores (Y). The sample mean of X is 12 hours with variance $s_{x}^{2}=9$. The mean of Y is 78 with variance $s_{y}^{2}=64$. The sum of products $\sum (X_{i}-\bar X)(Y_{i}-\bar Y)$ equals 180.

(a) Compute the Pearson correlation coefficient $r$.

(b) Determine the least‑squares regression line predicting Y from X in the form $\hat Y = a + bX$.

(c) At the 5 % significance level, test whether the slope $b$ differs from zero (use a two‑tailed t‑test with $n-2$ degrees of freedom). State the conclusion.

Accepted Answer

r≈0.153; regression line $\hat Y = 73.10 + 0.41X$; slope not significant at 5% (fail to reject H0).

City Guilds Model Questions Vol1 2022 Information Technology Question 15

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