Justifying CLT for confidence intervals

4 The masses, \(m\) kilograms, of flour in a random sample of 90 sacks of flour are summarised as follows. $$n = 90 \quad \Sigma m = 4509 \quad \Sigma m ^ { 2 } = 225950$$

1. Find unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.
3. Explain why it was necessary to use the Central Limit theorem in answering part (b).
4. Find the probability that the confidence interval found in part (b) is wholly above the true value of the population mean.

2 The widths, \(w \mathrm {~cm}\), of a random sample of 150 leaves of a certain kind were measured. The sample mean of \(w\) was found to be 3.12 cm . Using this sample, an approximate \(95 \%\) confidence interval for the population mean of the widths in centimetres was found to be [3.01, 3.23].

1. Calculate an estimate of the population standard deviation.
2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (a). [1]

1 A construction company notes the time, \(t\) days, that it takes to build each house of a certain design. The results for a random sample of 60 such houses are summarised as follows. $$\Sigma t = 4820 \quad \Sigma t ^ { 2 } = 392050$$

1. Calculate a 98\% confidence interval for the population mean time.
2. Explain why it was necessary to use the Central Limit theorem in part (a).

4 The lengths, \(x \mathrm {~m}\), of a random sample of 200 balls of string are found and the results are summarised by \(\Sigma x = 2005\) and \(\Sigma x ^ { 2 } = 20175\).

1. Calculate unbiased estimates of the population mean and variance of the lengths.
2. Use the values from part (i) to estimate the probability that the mean length of a random sample of 50 balls of string is less than 10 m .
3. Explain whether or not it was necessary to use the Central Limit theorem in your calculation in part (ii).

2 The standard deviation of the volume of drink in cans of Koola is 4.8 centilitres. A random sample of 180 cans is taken and the mean volume of drink in these 180 cans is found to be 330.1 centilitres.

1. Calculate a \(95 \%\) confidence interval for the mean volume of drink in all cans of Koola. Give the end-points of your interval correct to 1 decimal place.
2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).

1 The masses of a certain variety of plums are known to have standard deviation 13.2 g . A random sample of 200 of these plums is taken and the mean mass of the plums in the sample is found to be 62.3 g .

1. Calculate a \(98 \%\) confidence interval for the population mean mass.
2. State with a reason whether it was necessary to use the Central Limit theorem in the calculation in part (i).

4 Part of an ecological study involved measuring the heights of trees in a young forest. In order to obtain an estimate of the mean height of all the trees in the forest, a random sample of 70 trees was selected and their heights measured. These heights, \(x\) metres, are summarised by \(\Sigma x = 246.6\) and \(\Sigma x ^ { 2 } = 1183.65\). The mean height of all trees in the forest is denoted by \(\mu\) metres.

1. Calculate a symmetric \(90 \%\) confidence interval for \(\mu\).
2. A student was asked to interpret the interval and said,
   "If 100 independent \(90 \%\) confidence intervals were calculated then 90 of them would contain \(\mu\)." Explain briefly what is wrong with this statement.
3. Four independent \(90 \%\) confidence intervals for \(\mu\) are obtained. Calculate the probability that at least three of the intervals contain \(\mu\).

4 A random sample of 160 observations of a random variable \(X\) is selected. The sample can be summarised as follows. \(n = 160 \quad \sum x = 2688 \quad \sum x ^ { 2 } = 48398\)

1. Calculate unbiased estimates of the following.
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\)
2. Find a 99\% confidence interval for \(\mathrm { E } ( X )\), giving the end-points of the interval correct to 4 significant figures.
3. Explain whether it was necessary to use the Central Limit Theorem in answering
   1. part (a),
   2. part (b).

3. (a) Explain what you understand by the Central Limit Theorem. A garage services hire cars on behalf of a hire company. The garage knows that the lifetime of the brake pads has a standard deviation of 5000 miles. The garage records the lifetimes, \(x\) miles, of the brake pads it has replaced. The garage takes a random sample of 100 brake pads and finds that \(\sum x = 1740000\) (b) Find a 95\% confidence interval for the mean lifetime of a brake pad.
(c) Explain the relevance of the Central Limit Theorem in part (b). Brake pads are made to be changed every 20000 miles on average.
The hire car company complain that the garage is changing the brake pads too soon.
(d) Comment on the hire company's complaint. Give a reason for your answer.

6 The table below shows the mean and variance of the test scores of a random samples of 70 girls who are starting an A level Mathematics course.

1. Showing your working, find a \(95 \%\) confidence interval for the population mean.
2. Explain why you can construct the interval in part (i) despite no information about the distribution of the parent population being given.
3. The same random sample of girls repeats the test. The mean improvement in score is 0.9 . The \(95 \%\) confidence interval for the improvement is \([ - 1.5,3.3 ]\). What is the sample variance for the improvement in score?

1 When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumference is 34.711 cm . The sample standard deviation head circumference is 1.530 cm .

1. Determine a 95\% confidence interval for the population mean head circumference of newborn female babies.
2. Explain why you can calculate this interval even though the distribution of the population of head circumferences of newborn female babies is unknown.

3 The mass of male giraffes is assumed to have a normal distribution. Duncan takes a random sample of 600 male giraffes.
The mean mass of the sample is 1196 kilograms.
The standard deviation of the sample is 98 kilograms.
3

1. Construct a 94\% confidence interval for the mean mass of male giraffes, giving your values to one decimal place.
   3
2. Explain whether or not your answer to part (a) would change if a sample of size 5 was taken with the same mean and standard deviation.