Moderate -0.3 This is a straightforward one-tail z-test with all information provided directly: population parameters given, clear hypotheses, standard test procedure. Requires only routine application of the z-test formula and comparison with critical value. Slightly easier than average due to explicit setup and no conceptual complications, though the 2.5% significance level (rather than 5% or 10%) adds minor complexity.
1 In Europe the diameters of women's rings have mean 18.5 mm . Researchers claim that women in Jakarta have smaller fingers than women in Europe. The researchers took a random sample of 20 women in Jakarta and measured the diameters of their rings. The mean diameter was found to be 18.1 mm . Assuming that the diameters of women's rings in Jakarta have a normal distribution with standard deviation 1.1 mm , carry out a hypothesis test at the \(2 \frac { 1 } { 2 } \%\) level to determine whether the researchers' claim is justified.
For correct \(z\); Correct comparison with correct CV or finding area on LHS of \(-1.626\) and comparing with 2.5% (OR comparison with 2.241 if one-tail test set up)
Not enough evidence to support the claim that fingers are smaller
A1ft
Correct conclusion must fit their CV and their \(z\). No contradictions
[5]
$H_0 : \mu = 18.5$, $H_1 : \mu < 18.5$ | B1 | Both hypotheses correct
$z = \frac{18.1 - 18.5}{(1.1/\sqrt{20})} = -1.626$ | M1 | Standardising, must have $\sqrt{20}$
$CV = z = \pm 1.96$ | A1, M1 | For correct $z$; Correct comparison with correct CV or finding area on LHS of $-1.626$ and comparing with 2.5% (OR comparison with 2.241 if one-tail test set up)
Not enough evidence to support the claim that fingers are smaller | A1ft | Correct conclusion must fit their CV and their $z$. No contradictions
**[5]**
1 In Europe the diameters of women's rings have mean 18.5 mm . Researchers claim that women in Jakarta have smaller fingers than women in Europe. The researchers took a random sample of 20 women in Jakarta and measured the diameters of their rings. The mean diameter was found to be 18.1 mm . Assuming that the diameters of women's rings in Jakarta have a normal distribution with standard deviation 1.1 mm , carry out a hypothesis test at the $2 \frac { 1 } { 2 } \%$ level to determine whether the researchers' claim is justified.
\hfill \mbox{\textit{CAIE S2 2009 Q1 [5]}}