Moderate -0.3 This is a straightforward one-sample z-test with clearly stated hypotheses, given summary statistics, and a standard significance level. The calculation requires finding the sample mean, computing a z-statistic (assuming known population variance from the sample), and comparing to critical values. While it's a Further Maths topic, the execution is mechanical with no conceptual subtleties or multi-step problem-solving required, making it slightly easier than average.
7 Sweet pea plants grown using a standard plant food have a mean height of 1.6 m . A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following.
$$\begin{aligned}
n & = 49 \\
\Sigma x & = 74.48 \\
\Sigma x ^ { 2 } & = 120.8896
\end{aligned}$$
Test, at the \(5 \%\) significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m .
Question 7:
7 | 2 | 1 | 17 | 11 | 14 | 4
7 | (cid:80)(cid:32)x (cid:32)1.52
49(cid:167)120.8896 (cid:183)
(cid:86)ˆ2 (cid:32) (cid:168) (cid:16)1.522 (cid:184)
48(cid:169) 49 (cid:185)
(cid:32)0.16
H :(cid:80)(cid:32)1.6
0
H :(cid:80)(cid:31)1.6
1
1.52(cid:16)1.6
p(cid:32)0.0808 or z(cid:32) (cid:32)(cid:16)1.4
0.16
49
0.0808(cid:33)0.05or (cid:16)1.4(cid:33)(cid:16)1.645
Do not reject H
0
Insufficient evidence that height of plants using
new plant food is less than 1.6
p
S | B1
B1
B1
B1
M1
A1
A1
c
M1
eA1FT
[9] | 3.1b
3.3
1.1
2.5
2.1
3.4
i
1.1
1.1
2.2b | 1.52 seen
Biased estimate (0.1567) B0 but can
get all subsequent marks
Hypotheses both correct, B2. One
n
error, B1, but use of x orxor 1.52 is
B0B0
e
Evidence for 49 divisor needed (see
notes)
m
p(cid:32)0.0808or z(cid:32)(cid:16)1.4seen, allow
(cid:14)1.4
BC
Allow 1.4(cid:31)1.645 only if consistent
Correct method, comparison and
conclusion
Contextualised, acknowledge
uncertainty, needs double negative [not
“evidence that height is 1.6”]. FT on z.
Do not award final M1A1 if either 49
divisor missing or hypotheses given in
terms of 1.52 | ((cid:68)) Unless wrong working is
seen, p(cid:32)0.0808 or z(cid:32)(cid:16)1.4
automatically gets M1A1 and
(unless hypotheses are given in
terms of 1.52) automatically
qualifies for A1M1A1FT
((cid:69)) If neither p(cid:32)0.0808 or
z(cid:32)(cid:16)1.4 is seen, all of the last
5 marks depend on seeing
(cid:167) 0.16(cid:183)
either N 1.6, oe, or
(cid:168) (cid:184)
(cid:169) 49 (cid:185)
1.52(cid:16)1.6
. Either of these
0.16
49
seen but with square root
errors can get
M1A0A1M1A1FT
((cid:74)) “cdfnorm” notation, or
similar, with wrong p or z does
not qualify for M1A0A1 but
can get last M1A1FT provided
49 is seen to be used and
hypotheses not stated in terms
of 1.52. “cdfnorm” notation
with correct p or z can get full
marks.
7 | 3 | 3 | 1 | 29 | m
7 Sweet pea plants grown using a standard plant food have a mean height of 1.6 m . A new plant food is used for a random sample of 49 randomly chosen plants and the heights, $x$ metres, of this sample can be summarised by the following.
$$\begin{aligned}
n & = 49 \\
\Sigma x & = 74.48 \\
\Sigma x ^ { 2 } & = 120.8896
\end{aligned}$$
Test, at the $5 \%$ significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m .
\hfill \mbox{\textit{OCR Further Statistics Q7 [9]}}