Confidence interval from coded data

2 Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, \(x \mathrm {~m}\), for each kangaroo. He then calculates a \(90 \%\) confidence interval for the population mean tail length, \(\mu \mathrm { m }\), of male red kangaroos. He assumes that the tail lengths are normally distributed and finds that \(1.11 \leqslant \mu \leqslant 1.14\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.

2 A rowing club has a large number of members.A random sample of 12 of these members is taken and the pulse rate,\(x\) beats per minute(bpm),of each is measured after a 30 -minute training session.A \(98 \%\) confidence interval for the population mean pulse rate,\(\mu \mathrm { bpm }\) ,is calculated from the sample as \(64.22 < \mu < 68.66\) .

1. Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) .
2. State an assumption that is necessary for the confidence interval to be valid. \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-04_2718_38_141_2009}

6 Nassa is researching the lengths of a particular type of snake in two countries, \(A\) and \(B\).

1. He takes a random sample of 10 snakes of this type from country \(A\) and measures the length, \(x \mathrm {~m}\), of each snake. He then calculates a \(90 \%\) confidence interval for the population mean length, \(\mu \mathrm { m }\), for snakes of this type, assuming that snake lengths have a normal distribution. This confidence interval is \(3.36 \leqslant \mu \leqslant 4.22\). Find the sample mean and an unbiased estimate for the population variance.
2. Nassa also measures the lengths, \(y \mathrm {~m}\), of a random sample of 8 snakes of this type taken from country \(B\). His results are summarised as follows. $$\sum y = 27.86 \quad \sum y ^ { 2 } = 98.02$$ Nassa claims that the mean length of snakes of this type in country \(B\) is less than the mean length of snakes of this type in country \(A\). Nassa assumes that his sample from country \(B\) also comes from a normal distribution, with the same variance as the distribution from country \(A\). Test at the \(10 \%\) significance level whether there is evidence to support Nassa’s claim.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 A basketball club has a large number of players. The heights, \(x \mathrm {~m}\), of a random sample of 10 of these players are measured. A \(90 \%\) confidence interval for the population mean height, \(\mu \mathrm { m }\), of players in this club is calculated. It is assumed that heights are normally distributed. The confidence interval is \(1.78 \leqslant \mu \leqslant 2.02\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.

3 A random sample of 80 precision-engineered cylindrical components is checked as part of a quality control process. The diameters of the cylinders should be 25.00 cm . Accurate measurements of the diameters, \(x \mathrm {~cm}\), for the sample are summarised by $$\Sigma ( x - 25 ) = 0.44 , \quad \Sigma ( x - 25 ) ^ { 2 } = 0.2287 .$$

1. Calculate a \(99 \%\) confidence interval for the population mean diameter of the components.
2. For the calculation in part (i) to be valid, is it necessary to assume that component diameters are normally distributed? Justify your answer.

1 A machine fills bottles with mineral water.
The machine is checked every day to ensure that it is working correctly. On a particular day a random sample of 100 bottles is taken. The volume of water, \(x\) millilitres, for each bottle is measured and each measurement is coded using $$y = x - 1000$$ The results are summarised below $$\sum y = 847 \quad \sum y ^ { 2 } = 13510.09$$

1. 1. Show that the value of the unbiased estimate of the mean of \(x\) is 1008.47
   2. Calculate the unbiased estimate of the variance of \(x\) The machine was initially set so that the volume of water in a bottle had a mean value of 1010 millilitres. Later, a test at the \(5 \%\) significance level is used to determine whether or not the mean volume of water in a bottle has changed. If it has changed then the machine is stopped and reset.
2. Write down suitable null and alternative hypotheses for a 2-tailed test.
3. Find the critical region for \(\bar { X }\) in the above test.
4. Using your answer to part (a) and your critical region found in part (c), comment on whether or not the machine needs to be stopped and reset.
   Give a reason for your answer.
5. Explain why the use of \(\sigma ^ { 2 } = s ^ { 2 }\) is reasonable in this situation.

7 A factory produces 3-litre bottles of mineral water. The volume of water in a bottle has previously had a mean value of 3020 millilitres. Following a stoppage for maintenance, the volume of water, \(x\) millilitres, in each of a random sample of 100 bottles is measured and the following data obtained, where \(y = x - 3000\). $$\sum y = 1847.0 \quad \sum ( y - \bar { y } ) ^ { 2 } = 6336.00$$

1. Carry out a hypothesis test, at the \(5 \%\) significance level, to investigate whether the mean volume of water in a bottle has changed.
   (8 marks)
2. Subsequent measurements establish that the mean volume of water in a bottle produced by the factory after the stoppage is 3020 millilitres. State whether a Type I error, a Type II error or no error was made when carrying out the test in part (a).
   (l mark)