Moderate -0.8 This is a straightforward one-tail z-test with all information provided directly: known population standard deviation, clear hypotheses, and standard significance level. It requires only routine application of the z-test formula and comparison with critical value—no conceptual challenges or multi-step reasoning beyond the standard hypothesis testing procedure.
3 The lengths, in centimetres, of rods produced in a factory have mean \(\mu\) and standard deviation 0.2. The value of \(\mu\) is supposed to be 250 , but a manager claims that one machine is producing rods that are too long on average. A random sample of 40 rods from this machine is taken and the sample mean length is found to be 250.06 cm . Test at the \(5 \%\) significance level whether the manager's claim is justified.
M1 for standardising, must have \(\sqrt{40}\). Accept cv method
\(= 1.90\)
A1
comp with \(z = 1.645\); Claim is justified or There is evidence that claim is true
M1 A1\(\checkmark\) [5]
For valid comparison "1.90" with 1.645 or area comparison or CVs. Correct conclusion. No contradictions. NB 2-tail test scores B0 M1 A1 M1 (use 1.96) A0
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu = 250$; $H_1: \mu > 250$ | B1 | Both hypotheses |
| $\frac{250.06 - 250}{0.2 \div \sqrt{40}}$ | M1 | M1 for standardising, must have $\sqrt{40}$. Accept cv method |
| $= 1.90$ | A1 | |
| comp with $z = 1.645$; Claim is justified or There is evidence that claim is true | M1 A1$\checkmark$ [5] | For valid comparison "1.90" with 1.645 or area comparison or CVs. Correct conclusion. No contradictions. NB 2-tail test scores B0 M1 A1 M1 (use 1.96) A0 |
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3 The lengths, in centimetres, of rods produced in a factory have mean $\mu$ and standard deviation 0.2. The value of $\mu$ is supposed to be 250 , but a manager claims that one machine is producing rods that are too long on average. A random sample of 40 rods from this machine is taken and the sample mean length is found to be 250.06 cm . Test at the $5 \%$ significance level whether the manager's claim is justified.
\hfill \mbox{\textit{CAIE S2 2014 Q3 [5]}}