Standard +0.8 This is a small-sample confidence interval requiring the t-distribution with manual calculation of sample mean and standard deviation from raw data (6 values), followed by a straightforward linear transformation. While conceptually standard for Further Maths statistics, the computational demands and use of t-tables make it moderately above average difficulty.
A machine produces metal discs whose diameters have a normal distribution. The mean of this distribution is intended to be \(10\) cm. Accuracy is checked by measuring the diameters of a random sample of six discs. The diameters, in cm, are as follows.
10.03 \quad 10.02 \quad 9.98 \quad 10.06 \quad 10.08 \quad 10.01
Calculate a 99\% confidence interval for the mean diameter of all discs produced by the machine. [5]
Deduce a 99\% confidence interval for the mean circumference of all discs produced by the machine. [1]
Find sample mean (2 dp required): x = 60⋅18 / 6 = 10⋅03 B1
Estimate population variance to 3 sf (allow biased here: 0⋅00107 or 0⋅0327 2 ):
s 2 = (603⋅6118 – 60⋅18 2 /6) / 5 = 0⋅00128 or 0⋅0358 2 B1
Use consistent formula for C.I. with any t value: x ± t√(s 2 /6) M1√
Use of correct tabular value: t5, 0.995 = 4⋅03[2] *A1
C.I. correct to 2 dp in cm (dep *A1): 10⋅03 ± 0⋅06 or [9⋅97, 10⋅09] A1
Find CI for mean circumference to 1 dp in cm: 10⋅03π ± 0⋅06π
Answer
Marks
= 31⋅5 ± 0⋅2 or [31⋅3, 31⋅7] B1
(5)
(1)
[6]
Question 6:
6 | Find sample mean (2 dp required): x = 60⋅18 / 6 = 10⋅03 B1
Estimate population variance to 3 sf (allow biased here: 0⋅00107 or 0⋅0327 2 ):
s 2 = (603⋅6118 – 60⋅18 2 /6) / 5 = 0⋅00128 or 0⋅0358 2 B1
Use consistent formula for C.I. with any t value: x ± t√(s 2 /6) M1√
Use of correct tabular value: t5, 0.995 = 4⋅03[2] *A1
C.I. correct to 2 dp in cm (dep *A1): 10⋅03 ± 0⋅06 or [9⋅97, 10⋅09] A1
Find CI for mean circumference to 1 dp in cm: 10⋅03π ± 0⋅06π
= 31⋅5 ± 0⋅2 or [31⋅3, 31⋅7] B1 | (5)
(1) | [6]
A machine produces metal discs whose diameters have a normal distribution. The mean of this distribution is intended to be $10$ cm. Accuracy is checked by measuring the diameters of a random sample of six discs. The diameters, in cm, are as follows.
10.03 \quad 10.02 \quad 9.98 \quad 10.06 \quad 10.08 \quad 10.01
Calculate a 99\% confidence interval for the mean diameter of all discs produced by the machine. [5]
Deduce a 99\% confidence interval for the mean circumference of all discs produced by the machine. [1]
\hfill \mbox{\textit{CAIE FP2 2009 Q6 [6]}}