Moderate -0.3 This is a straightforward two-tailed z-test with all information provided directly: known variance, clear hypotheses, and standard significance level. It requires only routine application of the z-test formula and comparison with critical values, making it slightly easier than average but still requiring proper statistical procedure.
3 A machine has produced nails over a long period of time, where the length in millimetres was distributed as \(\mathrm { N } ( 22.0,0.19 )\). It is believed that recently the mean length has changed. To test this belief a random sample of 8 nails is taken and the mean length is found to be 21.7 mm . Carry out a hypothesis test at the \(5 \%\) significance level to test whether the population mean has changed, assuming that the variance remains the same.
Under \(H_0\), test statistic \(z = \frac{21.7 - 22}{\sqrt{0.19/\sqrt{8}}} = -1.947\)
Critical value \(z = \pm 1.96\)
Answer
Marks
Guidance
Not in CR, not enough evidence of change.
B1, M1, A1, M1, A1†ℜ
Both correct, alternative hypothesis must be \(\neq\). Standardising, must see \(\sqrt{8}\) in denom. Correct test statistic \(\pm\) (accept rounding to 1.95). Comparison with correct CV must be \(\pm 1.96\) (or z consistent with \(H_1\)) or area comparison. Correct conclusion if their test statistic and CV (OR \(22±1.96\sqrt{(0.19/8)}\) or z consistent with \(H_1\) M1 A1†ℜ then comparison with 21.7 M1 A1ℜ).
$H_0: \mu = 22$
$H_1: \mu \neq 22$
Under $H_0$, test statistic $z = \frac{21.7 - 22}{\sqrt{0.19/\sqrt{8}}} = -1.947$
Critical value $z = \pm 1.96$
Not in CR, not enough evidence of change. | B1, M1, A1, M1, A1†ℜ | Both correct, alternative hypothesis must be $\neq$. Standardising, must see $\sqrt{8}$ in denom. Correct test statistic $\pm$ (accept rounding to 1.95). Comparison with correct CV must be $\pm 1.96$ (or z consistent with $H_1$) or area comparison. Correct conclusion if their test statistic and CV (OR $22±1.96\sqrt{(0.19/8)}$ or z consistent with $H_1$ M1 A1†ℜ then comparison with 21.7 M1 A1ℜ).
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3 A machine has produced nails over a long period of time, where the length in millimetres was distributed as $\mathrm { N } ( 22.0,0.19 )$. It is believed that recently the mean length has changed. To test this belief a random sample of 8 nails is taken and the mean length is found to be 21.7 mm . Carry out a hypothesis test at the $5 \%$ significance level to test whether the population mean has changed, assuming that the variance remains the same.
\hfill \mbox{\textit{CAIE S2 2007 Q3 [5]}}