CAIE
S2
2018
June
Q3
5 marks
Standard +0.3
3 A researcher wishes to estimate the proportion, \(p\), of houses in London Road that have only one occupant. He takes a random sample of 64 houses in London Road and finds that 8 houses in the sample have only one occupant. Using this sample, he calculates that an approximate \(\alpha \%\) confidence interval for \(p\) has width 0.130 . Find \(\alpha\) correct to the nearest integer.
CAIE
S2
2019
June
Q5
6 marks
Challenging +1.2
5 The amount of money, in dollars, spent by a customer on one visit to a certain shop is modelled by the distribution \(\mathrm { N } ( \mu , 1.94 )\). In the past, the value of \(\mu\) has been found to be 20.00 , but following a rearrangement in the shop, the manager suspects that the value of \(\mu\) has changed. He takes a random sample of 6 customers and notes how much they each spend, in dollars. The results are as follows.
15.50
17.60
17.30
22.00
23.50
31.00
The manager carries out a hypothesis test using a significance level of \(\alpha \%\). The test does not support his suspicion. Find the largest possible value of \(\alpha\).
CAIE
S2
2017
November
Q3
4 marks
Standard +0.3
3 After an election 153 adults, from a random sample of 200 adults, said that they had voted. Using this information, an \(\alpha \%\) confidence interval for the proportion of all adults who voted in the election was found to be 0.695 to 0.835 , both correct to 3 significant figures. Find the value of \(\alpha\), correct to the nearest integer.
CAIE
FP2
2009
June
Q6
6 marks
Standard +0.8
6 The times taken by employees in a factory to complete a certain task have a normal distribution with mean \(\mu\) seconds and standard deviation \(\sigma\) seconds, both of which are unknown. Based on a random sample of 20 employees, the symmetric \(95 \%\) confidence interval for \(\mu\) is \(( 481,509 )\). Calculate a symmetric \(90 \%\) confidence interval for \(\mu\).
[0pt]
[6]
CAIE
FP2
2015
June
Q7
7 marks
Challenging +1.2
7 A random sample of 8 sunflower plants is taken from the large number grown by a gardener, and the heights of the plants are measured. A 95\% confidence interval for the population mean, \(\mu\) metres, is calculated from the sample data as \(1.17 < \mu < 2.03\). Given that the height of a sunflower plant is denoted by \(x\) metres, find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\) for this sample of 8 plants.
CAIE
FP2
2016
June
Q11 OR
Challenging +1.8
Petra is studying a particular species of bird. She takes a random sample of 12 birds from nature reserve \(A\) and measures the wing span, \(x \mathrm {~cm}\), for each bird. She then calculates a \(95 \%\) confidence interval for the population mean wing span, \(\mu \mathrm { cm }\), for birds of this species, assuming that wing spans are normally distributed. Later, she is not able to find the summary of the results for the sample, but she knows that the \(95 \%\) confidence interval is \(25.17 \leqslant \mu \leqslant 26.83\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.
Petra also measures the wing spans of a random sample of 7 birds from nature reserve \(B\). Their wing spans, \(y \mathrm {~cm}\), are as follows.
$$\begin{array} { l l l l l l l }
23.2 & 22.4 & 27.6 & 25.3 & 28.4 & 26.5 & 23.6
\end{array}$$
She believes that the mean wing span of birds found in nature reserve \(A\) is greater than the mean wing span of birds found in nature reserve \(B\). Assuming that this second sample also comes from a normal distribution, with variance the same as the first distribution, test, at the \(10 \%\) significance level, whether there is evidence to support Petra's belief.