Standard unbiased estimates calculation

1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4
5.2
5.5
6.1

1. Calculate unbiased estimates of the population mean and variance of \(X\).
   It is now given that the true value of the population variance of \(X\) is 0.55 , and that \(X\) has a normal distribution.
2. Find a 95\% confidence interval for the population mean of \(X\).

1 A random sample of 100 values of a variable \(X\) is taken. These values are summarised below. $$n = 100 \quad \Sigma x = 1556 \quad \Sigma x ^ { 2 } = 29004$$ Calculate unbiased estimates of the population mean and variance of \(X\).

1 The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.
200
201
198
202
200
199
199
201
197
202
200
199

1. Find unbiased estimates of the population mean and variance.
2. Give a statistical reason why these estimates may not be reliable.

1 The weights, in grams, of a random sample of 8 packets of cereal are as follows. $$\begin{array} { l l l l l l l l } 250 & 248 & 255 & 244 & 259 & 250 & 242 & 258 \end{array}$$ Calculate unbiased estimates of the population mean and variance.

2 A researcher is investigating the lengths, in kilometres, of the journeys to work of the employees at a certain firm. She takes a random sample of 10 employees.

1. State what is meant by 'random' in this context. The results of her sample are as follows. $$\begin{array} { l l l l l l l l l l } 1.5 & 2.0 & 3.6 & 5.9 & 4.8 & 8.7 & 3.5 & 2.9 & 4.1 & 3.0 \end{array}$$
2. Find unbiased estimates of the population mean and variance.
3. State what is meant by 'population' in this context.

1 The heights, in metres, of a random sample of 10 mature trees of a certain variety are given below.
\(\begin{array} { l l l l l l l l l l } 5.9 & 6.5 & 6.7 & 5.9 & 6.9 & 6.0 & 6.4 & 6.2 & 5.8 & 5.8 \end{array}\) Find unbiased estimates of the population mean and variance of the heights of all mature trees of this variety.

3 The times, \(T\) minutes, taken by a random sample of 75 students to complete a test were noted. The results were summarised by \(\Sigma t = 230\) and \(\Sigma t ^ { 2 } = 930\).

1. Calculate unbiased estimates of the population mean and variance of \(T\).
   You should now assume that your estimates from part (a) are the true values of the population mean and variance of \(T\).
2. The times taken by another random sample of 75 students were noted, and the sample mean, \(\bar { T }\), was found. Find the value of \(a\) such that \(P ( \bar { T } > a ) = 0.234\).

4 Packets of cat food are filled by a machine.

1. In a random sample of 10 packets, the weights, in grams, of the packets were as follows.
   \(\begin{array} { l l l l l l l l l l } 374.6 & 377.4 & 376.1 & 379.2 & 371.2 & 375.0 & 372.4 & 378.6 & 377.1 & 371.5 \end{array}\)
   Find unbiased estimates of the population mean and variance.
2. In a random sample of 200 packets, 38 were found to be underweight. Calculate a \(96 \%\) confidence interval for the population proportion of underweight packets.

2 Jenny has to do a statistics project at school on how much pocket money, in dollars, is received by students in her year group. She plans to take a sample of 7 students from her year group, which contains 122 students.

1. Give a suitable method of taking this sample. Her sample gives the following results. $$\begin{array} { l l l l l l l } 13.40 & 10.60 & 26.50 & 20.00 & 14.50 & 15.00 & 16.50 \end{array}$$
2. Find unbiased estimates of the population mean and variance.
3. Is the estimated population variance more than, less than or the same as the sample variance?
4. Describe what you understand by 'population' in this question.

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 \(X\).

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 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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1 The weights, in kilograms, of a random sample of eight 16-year old males are given below. $$\begin{array} { l l l l l l l l } 58.9 & 63.5 & 62.7 & 59.4 & 66.9 & 68.0 & 60.4 & 68.2 \end{array}$$ Find unbiased estimates of the population mean and variance of the weights of all 16 -year old males.

4 A company sells sugar in bags which are labelled as containing 450 grams.
Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.

1. State two adjustments the company could make. The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
2. Given that \(\sum x = 11409\) and \(\sum x ^ { 2 } = 5206937\), calculate the sample mean and sample standard deviation of these weights.

2 A company sells sugar in bags which are labelled as containing 450 grams.
Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.

1. State two adjustments the company could make. The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
2. Given that \(\sum x = 11409\) and \(\sum x ^ { 2 } = 5206937\), calculate the sample mean and sample standard deviation of these weights.

1 A random sample of 50 observations of the random variable \(X\) is summarised by $$n = 50 , \Sigma x = 182.5 , \Sigma x ^ { 2 } = 739.625 .$$ Calculate unbiased estimates of the expectation and variance of \(X\).

1 The results of 14 observations of a random variable \(V\) are summarised by $$n = 14 , \quad \sum v = 3752 , \quad \sum v ^ { 2 } = 1007448 .$$ Calculate unbiased estimates of \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).

1 A random sample of nine observations of a random variable is obtained. The results are summarised as $$\Sigma x = 468 , \quad \Sigma x ^ { 2 } = 24820 .$$ Calculate unbiased estimates of the population mean and variance.

7. A random sample of 8 apples is taken from an orchard and the weight, in grams, of each apple is measured. The results are given below. $$\begin{array} { l l l l l l l l } 143 & 131 & 165 & 122 & 137 & 155 & 148 & 151 \end{array}$$

1. Calculate unbiased estimates for the mean and the variance of the weights of apples. A population has an unknown mean \(\mu\) and an unknown variance \(\sigma ^ { 2 }\)
   A random sample represented by \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 8 }\) is taken from this population.
2. Explain why \(\sum _ { i = 1 } ^ { 8 } \left( X _ { i } - \mu \right) ^ { 2 }\) is not a statistic. Given that \(\mathrm { E } \left( S ^ { 2 } \right) = \sigma ^ { 2 }\), where \(S ^ { 2 }\) is an unbiased estimator of \(\sigma ^ { 2 }\) and the statistic $$Y = \frac { 1 } { 8 } \left( \sum _ { i = 1 } ^ { 8 } X _ { i } ^ { 2 } - 8 \bar { X } ^ { 2 } \right)$$
3. find \(\mathrm { E } ( Y )\) in terms of \(\sigma ^ { 2 }\)
4. Hence find the bias, in terms of \(\sigma ^ { 2 }\), when \(Y\) is used as an estimator of \(\sigma ^ { 2 }\)
   

3 When an alarm is raised at a market town's fire station, the fire engine cannot leave until at least five fire-fighters arrive at the station. The call-out time, \(X\) minutes, is the time between an alarm being raised and the fire engine leaving the station. The value of \(X\) was recorded on a random sample of 50 occasions. The results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 286.5 \quad \sum ( x - \bar { x } ) ^ { 2 } = 45.16$$

1. Find values for the mean and standard deviation of this sample of 50 call-out times.
2. Hence construct a \(99 \%\) confidence interval for the mean call-out time.
3. The fire and rescue service claims that the station's mean call-out time is less than 5 minutes, whereas a parish councillor suggests that it is more than \(6 \frac { 1 } { 2 }\) minutes. Comment on each of these claims.

1 The values of 5 independent observations from a population can be summarised by $$\Sigma x = 75.8 , \quad \Sigma x ^ { 2 } = 1154.58 .$$ Find unbiased estimates of the population mean and variance.