Sample size determination

4 A new computer was bought by a local council to search council records and was tested by an employee. She searched a random sample of 500 records and the sample mean search time was found to be 2.18 milliseconds and an unbiased estimate of variance was \(1.58 ^ { 2 }\) milliseconds \({ } ^ { 2 }\).

1. Calculate a \(98 \%\) confidence interval for the population mean search time \(\mu\) milliseconds.
2. It is required to obtain a sample mean time that differs from \(\mu\) by less than 0.05 milliseconds with probability 0.95 . Estimate the sample size required.
3. State why it is unnecessary for the validity of your calculations that search time has a normal distribution.

3 The proportion, \(p\), of an island's population with blood type \(\mathrm { A } \mathrm { Rh } ^ { + }\)is believed to be approximately 0.35 . A medical organisation, requiring a more accurate estimate, specifies that a \(98 \%\) confidence interval for \(p\) should have a width of at most 0.1 . Calculate, to the nearest 10, an estimate of the minimum sample size necessary in order to achieve the organisation's requirement.

4
The waiting time at a hospital's A\&E department may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\frac { \mu } { 2 }\).
The department's manager wishes a \(95 \%\) confidence interval for \(\mu\) to be constructed such that it has a width of at most \(0.2 \mu\).
Calculate, to the nearest 10, an estimate of the minimum sample size necessary in order to achieve the manager's wish.
(5 marks)
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5 A random sample of 125 people was selected from a council's electoral roll. Of these, 68 were in favour of a proposed local building plan.

1. Construct an approximate 98\% confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal.
2. Calculate, to the nearest 5, an estimate of the minimum sample size necessary in order that an approximate \(98 \%\) confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal has a width of at most 10 per cent.

4 A machine is used to fill 5-litre plastic containers with vinegar. The volume, in litres, of vinegar in a container filled by the machine may be assumed to be normally distributed with mean \(\mu\) and standard deviation 0.08 . A quality control inspector requires a \(99 \%\) confidence interval for \(\mu\) to be constructed such that it has a width of at most 0.05 litres. Calculate, to the nearest 5, the sample size necessary in order to achieve the inspector's requirement.

4 The height of lilac trees, in metres, can be modelled by a normal distribution with variance 0.7 A random sample of \(n\) lilac trees is taken and used to construct a 99\% confidence interval for the population mean. This confidence interval is \(( 5.239,5.429 )\) 4

1. Find the value of \(n\) 4
2. Joey claims that the mean height of lilac trees is 5.3 metres.
   State, with a reason, whether the confidence interval supports Joey's claim.

4 Oscar is studying the daily maximum temperature in \({ } ^ { \circ } \mathrm { C }\) in a village during the month of June. He constructs a \(95 \%\) confidence interval of width \(0.8 ^ { \circ } \mathrm { C }\) using a random sample of 150 days. He assumes that the daily maximum temperature has a normal distribution.
4

1. Find the standard deviation of Oscar's sample, giving your answer to three significant figures.
   4
2. Oscar calculates the mean of his sample to be \(25.3 ^ { \circ } \mathrm { C }\) He claims that the population mean is \(26.0 ^ { \circ } \mathrm { C }\) Explain whether or not his confidence interval supports his claim.
   4
3. Explain how Oscar could reduce the width of his 95\% confidence interval.