Standard +0.3 This is a straightforward one-sample t-test with clearly stated hypotheses (μ > 14.8), given summary statistics requiring calculation of sample mean and standard deviation, then comparison with critical value from tables. While it's a Further Maths topic, the execution is mechanical with no conceptual challenges beyond standard procedure.
Farmer A grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer B grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer A's trees. The masses of apples from Farmer B's trees may be assumed to be normally distributed.
A random sample of 10 trees from Farmer B is chosen. The masses, \(x\) kg, of apples produced in a year are summarised as follows.
$$\sum x = 152.0 \qquad \sum x^2 = 2313.0$$
Test, at the 5% significance level, whether Farmer B's claim is justified. [6]
Correct conclusion, in context, following correct work. Level
of uncertainty in language is used.
Allow ±1 difference in third significant figure of their t for this
mark.
6
Answer
Marks
Guidance
Question
Answer
Marks
Question 1:
1 | 1 1522
s2 = 2313− =0.28889
9 10
| M1 | 13
, accept unsimplified.
45
H : μ = 14.8 H : μ > 14.8
0 1 | B1 | If μ not used, ‘population mean’ required.
Must see 14.8, must be = and >.
152
−14.8
[ t= ] 10
s/ 10 | M1 | Using unbiased estimate.
p-value is 0.0215.
[ ]
t= 2.35 | A1 | Rounds to 2.35.
Tabular value = 1.833
‘2.35’ > 1.833
Reject H
0 | M1 | Comparison with 1.833 and correct FT conclusion.
Sufficient evidence to accept Farmer B’s claim oe | A1 | Correct conclusion, in context, following correct work. Level
of uncertainty in language is used.
Allow ±1 difference in third significant figure of their t for this
mark.
6
Question | Answer | Marks | Guidance
Farmer A grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer B grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer A's trees. The masses of apples from Farmer B's trees may be assumed to be normally distributed.
A random sample of 10 trees from Farmer B is chosen. The masses, $x$ kg, of apples produced in a year are summarised as follows.
$$\sum x = 152.0 \qquad \sum x^2 = 2313.0$$
Test, at the 5% significance level, whether Farmer B's claim is justified. [6]
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q1 [6]}}