5.05b Unbiased estimates: of population mean and variance

5 The volumes, \(v\) millilitres, of juice in a random sample of 50 bottles of Cooljoos are measured and summarised as follows. $$n = 50 \quad \Sigma v = 14800 \quad \Sigma v ^ { 2 } = 4390000$$

1. Find unbiased estimates of the population mean and variance.
2. An \(\alpha \%\) confidence interval for the population mean, based on this sample, is found to have a width of 5.45 millilitres. Find \(\alpha\). Four random samples of size 10 are taken and a \(96 \%\) confidence interval for the population mean is found from each sample.
3. Find the probability that these 4 confidence intervals all include the true value of the population mean.

1 A random sample of 75 values of a variable \(X\) gave the following results. $$n = 75 \quad \Sigma x = 153.2 \quad \Sigma x ^ { 2 } = 340.24$$ Find unbiased estimates for the population mean and variance of \(X\).

4 The mean mass of packets of sugar is supposed to be 505 g . A random sample of 10 packets filled by a certain machine was taken and the masses, in grams, were found to be as follows. $$\begin{array} { l l l l l l l l l l } 500 & 499 & 496 & 495 & 498 & 490 & 492 & 501 & 494 & 494 \end{array}$$

1. Find unbiased estimates of the population mean and variance.
   The mean mass of packets produced by this machine was found to be less than 505 g , so the machine was adjusted. Following the adjustment, the masses of a random sample of 150 packets from the machine were measured and the total mass was found to be 75660 g .
2. Given that the population standard deviation is 3.6 g , test at the \(2 \%\) significance level whether the machine is still producing packets with mean mass less than 505 g .
3. Explain why the use of the normal distribution is justified in carrying out the test in part (ii). [1]

3 It is claimed that, on average, a particular train journey takes less than 1.9 hours. The times, \(t\) hours, taken for this journey on a random sample of 50 days were recorded. The results are summarised below. $$n = 50 \quad \Sigma t = 92.5 \quad \Sigma t ^ { 2 } = 175.25$$

1. Calculate unbiased estimates of the population mean and variance.
2. Test the claim at the \(5 \%\) significance level.

2 The length of worms is denoted by \(X \mathrm {~cm}\). The lengths of a random sample of 50 worms were measured. Some of the results were lost, but the following results are available.

• \(\Sigma x ^ { 2 } = 4361\)
• An unbiased estimate of the population variance of \(X\) is 9.62.

Calculate the mean length of the 50 worms.

1 The lengths, \(X \mathrm {~cm}\), of a sample of 100 insects of a certain type were summarised as follows. $$n = 100 \quad \sum x = 36.8 \quad \sum x ^ { 2 } = 17.34$$

1. Calculate unbiased estimates for the population mean and variance of \(X\).
2. State a necessary condition for the estimates found in part (a) to be reliable.

4 The areas, \(X \mathrm {~cm} ^ { 2 }\), of petals of a certain kind of flower have mean \(\mu \mathrm { cm } ^ { 2 }\). In the past it has been found that \(\mu = 8.9\). Following a change in the climate, a botanist claims that the mean is no longer 8.9. The areas of a random sample of 200 petals from this kind of flower are measured, and the results are summarized by $$\Sigma x = 1850 , \quad \Sigma x ^ { 2 } = 17850 .$$ Test the botanist's claim at the \(2.5 \%\) significance level.

3 The masses, \(m \mathrm {~kg}\), of packets of flour are normally distributed. The mean mass is supposed to be 1.01 kg . A quality control officer measures the masses of a random sample of 100 packets. The results are summarised below. $$n = 100 \quad \Sigma m = 98.2 \quad \Sigma m ^ { 2 } = 104.52$$

1. Test at the \(5 \%\) significance level whether the population mean mass is less than 1.01 kg .
2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).

2 A headteacher models the number of children who bring a 'healthy' packed lunch to school on any day by the distribution \(\mathrm { B } ( 150 , p )\). In the past, she has found that \(p = \frac { 1 } { 3 }\). Following the opening of a fast food outlet near the school, she wishes to test, at the \(1 \%\) significance level, whether the value of \(p\) has decreased.

1. State the null and alternative hypotheses for this test.
   On a randomly chosen day she notes the number, \(N\), of children who bring a 'healthy' packed lunch to school. She finds that \(N = 36\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \leqslant 36 ) = 0.0084\).
2. State, with a reason, the conclusion that the headteacher should draw from the test.
3. According to the model, what is the largest number of children who might bring a packed lunch to school?

7 A mill owner claims that the mean mass of sacks of flour produced at his mill is 51 kg . A quality control officer suspects that the mean mass is actually less than 51 kg . In order to test the owner's claim she finds the mass, \(x \mathrm {~kg}\), of each of a random sample of 150 sacks and her results are summarised as follows. $$n = 150 \quad \Sigma x = 7480 \quad \Sigma x ^ { 2 } = 380000$$

1. Carry out the test at the \(2.5 \%\) significance level.
   You may now assume that the population standard deviation of the masses of sacks of flour is 6.856 kg . The quality control officer weighs another random sample of 150 sacks and carries out another test at the 2.5\% significance level.
2. Given that the population mean mass is 49 kg , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The numbers of basketball courts in a random sample of 70 schools in South Mowland are summarised in the table.

1. Calculate unbiased estimates for the population mean and variance of the number of basketball courts per school in South Mowland.
   The mean number of basketball courts per school in North Mowland is 1.9 .
2. Test at the \(5 \%\) significance level whether the mean number of basketball courts per school in South Mowland is less than the mean for North Mowland.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (ii).

6 In the past, Angus found that his train was late on \(15 \%\) of his daily journeys to work. Following a timetable change, Angus found that out of 60 randomly chosen days, his train was late on 6 days.

1. Test at the \(10 \%\) significance level whether Angus' train is late less often than it was before the timetable change.
   Angus used his random sample to find an \(\alpha \%\) confidence interval for the proportion of days on which his train is late. The upper limit of his interval was 0.150 , correct to 3 significant figures.
2. Calculate the value of \(\alpha\) correct to the nearest integer.

3 The masses, in grams, of bags of flour are normally distributed with mean \(\mu\). The masses, \(m\) grams, of a random sample of 50 bags are summarised by \(\Sigma m = 25110\) and \(\Sigma m ^ { 2 } = 12610300\).

1. Calculate a \(96 \%\) confidence interval for \(\mu\), giving the end-points correct to 1 decimal place.
   Another random sample of 50 bags of flour is taken and a \(99 \%\) confidence interval for \(\mu\) is calculated.
2. Without calculation, state whether this confidence interval will be wider or narrower than the confidence interval found in part (i). Give a reason for your answer.

3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below.

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.

4 The lifetimes, \(X\) hours, of a random sample of 50 batteries of a certain kind were found. The results are summarised by \(\Sigma x = 420\) and \(\Sigma x ^ { 2 } = 27530\).

1. Calculate an unbiased estimate of the population mean of \(X\) and show that an unbiased estimate of the population variance is 490 , correct to 3 significant figures.
2. The lifetimes of a further large sample of \(n\) batteries of this kind were noted, and the sample mean, \(\bar { X }\), was found. Use your estimates from part (i) to find the value of \(n\) such that \(\mathrm { P } ( \bar { X } > 5 ) = 0.9377\).
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3 A random sample of 150 students attending a college is taken, and their travel times, \(t\) minutes, are measured. The data are summarised by \(\Sigma t = 4080\) and \(\Sigma t ^ { 2 } = 159252\).

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(94 \%\) confidence interval for the population mean travel time.

7 In a research laboratory where plants are studied, the probability of a certain type of plant surviving was 0.35 . The laboratory manager changed the growing conditions and wished to test whether the probability of a plant surviving had increased.

1. The plants were grown in rows, and when the manager requested a random sample of 8 plants to be taken, the technician took all 8 plants from the front row. Explain what was wrong with the technician's sample.
2. A suitable sample of 8 plants was taken and 4 of these 8 plants survived. State whether the manager's test is one-tailed or two-tailed and also state the null and alternative hypotheses. Using a \(5 \%\) significance level, find the critical region and carry out the test.
3. State the meaning of a Type II error in the context of the test in part (ii).
4. Find the probability of a Type II error for the test in part (ii) if the probability of a plant surviving is now 0.4.

4 Diameters of golf balls are known to be normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm }\). A random sample of 130 golf balls was taken and the diameters, \(x \mathrm {~cm}\), were measured. The results are summarised by \(\Sigma x = 555.1\) and \(\Sigma x ^ { 2 } = 2371.30\).

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Calculate a \(97 \%\) confidence interval for \(\mu\).
3. 300 random samples of 130 balls are taken and a \(97 \%\) confidence interval is calculated for each sample. How many of these intervals would you expect not to contain \(\mu\) ?

6 Photographers often need to take many photographs of families until they find a photograph which everyone in the family likes. The number of photographs taken until obtaining one which everybody likes has mean 15.2. A new photographer claims that she can obtain a photograph which everybody likes with fewer photographs taken. To test at the \(10 \%\) level of significance whether this claim is justified, the numbers of photographs, \(x\), taken by the new photographer with a random sample of 60 families are recorded. The results are summarised by \(\Sigma x = 890\) and \(\Sigma x ^ { 2 } = 13780\).

1. Calculate unbiased estimates of the population mean and variance of the number of photographs taken by the new photographer.
2. State null and alternative hypotheses for the test, and state also the probability that the test results in a Type I error. Say what a Type I error means in the context of the question.
3. Carry out the test.

7

1. Give a reason why sampling would be required in order to reach a conclusion about
   1. the mean height of adult males in England,
   2. the mean weight that can be supported by a single cable of a certain type without the cable breaking.
2. The weights, in kg , of sacks of potatoes are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The weights of a random sample of 500 sacks of potatoes are found and the results are summarised below. $$n = 500 , \quad \Sigma x = 9850 , \quad \Sigma x ^ { 2 } = 194125 .$$
   1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   2. A further random sample of 60 sacks of potatoes is taken. Using your values from part (b) (i), find the probability that the mean weight of this sample exceeds 19.73 kg .
   3. Explain whether it was necessary to use the Central Limit Theorem in your calculation in part (b) (ii).

6 A clinic monitors the amount, \(X\) milligrams per litre, of a certain chemical in the blood stream of patients. For patients who are taking drug \(A\), it has been found that the mean value of \(X\) is 0.336 . A random sample of 100 patients taking a new drug, \(B\), was selected and the values of \(X\) were found. The results are summarised below. $$n = 100 , \quad \Sigma x = 43.5 , \quad \Sigma x ^ { 2 } = 31.56 .$$

1. Test at the \(1 \%\) significance level whether the mean amount of the chemical in the blood stream of patients taking drug \(B\) is different from that of patients taking drug \(A\).
2. For the test to be valid, is it necessary to assume a normal distribution for the amount of chemical in the blood stream of patients taking drug \(B\) ? Justify your answer.

4 The volumes of juice in bottles of Apricola are normally distributed. In a random sample of 8 bottles, the volumes of juice, in millilitres, were found to be as follows. $$\begin{array} { l l l l l l l l } 332 & 334 & 330 & 328 & 331 & 332 & 329 & 333 \end{array}$$

1. Find unbiased estimates of the population mean and variance. A random sample of 50 bottles of Apricola gave unbiased estimates of 331 millilitres and 4.20 millilitres \({ } ^ { 2 }\) for the population mean and variance respectively.
2. Use this sample of size 50 to calculate a \(98 \%\) confidence interval for the population mean.
3. The manufacturer claims that the mean volume of juice in all bottles is 333 millilitres. State, with a reason, whether your answer to part (ii) supports this claim.

5 It is claimed that, on average, people following the Losefast diet will lose more than 2 kg per month. The weight losses, \(x\) kilograms per month, of a random sample of 200 people following the Losefast diet were recorded and summarised as follows. $$n = 200 \quad \Sigma x = 460 \quad \Sigma x ^ { 2 } = 1636$$

1. Calculate unbiased estimates of the population mean and variance.
2. Test the claim at the \(1 \%\) significance level.

3 Following a change in flight schedules, an airline pilot wished to test whether the mean distance that he flies in a week has changed. He noted the distances, \(x \mathrm {~km}\), that he flew in 50 randomly chosen weeks and summarised the results as follows. $$n = 50 \quad \Sigma x = 143300 \quad \Sigma x ^ { 2 } = 410900000$$

1. Calculate unbiased estimates of the population mean and variance.
2. In the past, the mean distance that he flew in a week was 2850 km . Test, at the \(5 \%\) significance level, whether the mean distance has changed.

1 A random sample of 80 values of a variable \(X\) is taken and these values are summarised below. $$n = 80 \quad \Sigma x = 150.2 \quad \Sigma x ^ { 2 } = 820.24$$ Calculate unbiased estimates of the population mean and variance of \(X\) and hence find a \(95 \%\) confidence interval for the population mean of \(X\).