Standard +0.3 This is a standard application of the Mann-Whitney U test with clear hypotheses, straightforward ranking of combined data, calculation of U statistics, and comparison with critical values from tables. While it requires careful execution of the procedure, it involves no conceptual difficulty beyond following the standard algorithm for a non-parametric test, making it slightly easier than average for Further Maths statistics.
Nathan believes that shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. He takes a random sample of 8 shearers from Wales and 7 shearers from New Zealand. The numbers below indicate how many sheep were sheared in 45 minutes by the 15 shearers.
Wales: \quad 60 \quad 53 \quad 42 \quad 38 \quad 37 \quad 36 \quad 31 \quad 28
New Zealand: \quad 39 \quad 35 \quad 27 \quad 26 \quad 17 \quad 16 \quad 15
Use a Mann-Whitney U test at the 1\% significance level to test whether Nathan is correct. You must state your hypotheses clearly and state the critical region. [7]
\(H_0\): The median number of sheep sheared by shearers from Wales and New Zealand is the same.
—
Accept \(H_0: m_1 = m_2\) or \(H_1: m_1 > m_2\)
\(H_1\): The median number of sheep sheared by shearers from Wales is more than the median number of sheep sheared by shearers from New Zealand.
B1
—
Upper critical value is 48 OR Lower CV is 8
B1
—
The critical region is (\(U \geq 48\)) OR Critical region is (\(U \leq 8\))
B1
—
Use of the formula \(U = \sum\sum z_{ij}\)
M1
—
\(U = 7 + 7 + 7 + 6 + 6 + 6 + 5 + 5 = 49\)
A1
—
OR \(U = 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 = 7\)
—
—
49 is in the critical region OR 7 is in the critical region. There is sufficient evidence to reject \(H_0\).
B1
—
There is sufficient evidence to suggest shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand.
B1
—
Total: [7]
$H_0$: The median number of sheep sheared by shearers from Wales and New Zealand is the same. | — | Accept $H_0: m_1 = m_2$ or $H_1: m_1 > m_2$
$H_1$: The median number of sheep sheared by shearers from Wales is more than the median number of sheep sheared by shearers from New Zealand. | B1 | —
Upper critical value is 48 OR Lower CV is 8 | B1 | —
The critical region is ($U \geq 48$) OR Critical region is ($U \leq 8$) | B1 | —
Use of the formula $U = \sum\sum z_{ij}$ | M1 | —
$U = 7 + 7 + 7 + 6 + 6 + 6 + 5 + 5 = 49$ | A1 | —
OR $U = 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 = 7$ | — | —
49 is in the critical region OR 7 is in the critical region. There is sufficient evidence to reject $H_0$. | B1 | —
There is sufficient evidence to suggest shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. | B1 | —
**Total: [7]**
---
Nathan believes that shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. He takes a random sample of 8 shearers from Wales and 7 shearers from New Zealand. The numbers below indicate how many sheep were sheared in 45 minutes by the 15 shearers.
Wales: \quad 60 \quad 53 \quad 42 \quad 38 \quad 37 \quad 36 \quad 31 \quad 28
New Zealand: \quad 39 \quad 35 \quad 27 \quad 26 \quad 17 \quad 16 \quad 15
Use a Mann-Whitney U test at the 1\% significance level to test whether Nathan is correct. You must state your hypotheses clearly and state the critical region. [7]
\hfill \mbox{\textit{WJEC Further Unit 5 2019 Q7 [7]}}