8 An examination involved writing an essay. In order to compare the time taken to write the essay by students in two large colleges, a sample of 12 students from college \(A\) and a sample of 8 students from college \(B\) were randomly selected. The times, \(t _ { A }\) and \(t _ { B }\), taken for these students to write the essay were measured, correct to the nearest minute, and are summarised by $$n _ { A } = 12 , \quad \Sigma t _ { A } = 257 , \quad \Sigma t _ { A } ^ { 2 } = 5629 , \quad n _ { B } = 8 , \quad \Sigma t _ { B } = 206 , \quad \Sigma t _ { B } ^ { 2 } = 5359$$ Stating any required assumptions, calculate a \(95 \%\) confidence interval for the difference in the population means. State, giving a reason, whether your confidence interval supports the statement that the population means, for the two colleges, are equal.

## Question 8:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Equal variances | B1 | State assumptions (A.E.F.) |
| Normal populations | B1 | |
| $\bar{x}_A - \bar{x}_B = 21.417 - 25.75 = -4.33$ | B1 | Find difference in sample means to 2 dp |
| $s^2 = (5629 - 257^2/12 + 5359 - 206^2/18)/18 = (124.9 + 54.5)/18 = 9.968$ or $3.157^2$ (3 sf) | M1 A1 | Estimate common population variance |
| $t_{18,\,0.975} = 2.101$ (to 3 sf) | B1 | Use correct tabular $t$ value |
| $x_A - x_B \pm ts\sqrt{1/12 + 1/8}$ | M1 | Find confidence interval for $\bar{\mu}_A - \bar{\mu}_B$ |
| $-4.33 \pm 3.03$ or $[-7.36, -1.31]$ | A1* | Evaluate |
| Interval does not include zero so statement not supported | B1 | State reason and conclusion (A.E.F.) (dep *A1 apart from rounding) |

**Total: 9 marks**

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8 An examination involved writing an essay. In order to compare the time taken to write the essay by students in two large colleges, a sample of 12 students from college $A$ and a sample of 8 students from college $B$ were randomly selected. The times, $t _ { A }$ and $t _ { B }$, taken for these students to write the essay were measured, correct to the nearest minute, and are summarised by

$$n _ { A } = 12 , \quad \Sigma t _ { A } = 257 , \quad \Sigma t _ { A } ^ { 2 } = 5629 , \quad n _ { B } = 8 , \quad \Sigma t _ { B } = 206 , \quad \Sigma t _ { B } ^ { 2 } = 5359$$

Stating any required assumptions, calculate a $95 \%$ confidence interval for the difference in the population means.

State, giving a reason, whether your confidence interval supports the statement that the population means, for the two colleges, are equal.

\hfill \mbox{\textit{CAIE FP2 2010 Q8 [9]}}