Forward transformation: find new statistics

4 The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.

1. The mean cost of Fei's rides is \(
   ) 2.50\( and the standard deviation of the costs of Fei's rides is \)\\( 0\). Explain how you can tell that the roller coaster and the water slide each cost \(
   ) 2.50\( per ride. [2]
2. The mean cost of Graeme's rides is \)\\( 3.76\). Find the standard deviation of the costs of Graeme's rides.

4 The marks, \(x\), of a random sample of 50 students in a test were summarised as follows. $$n = 50 \quad \Sigma x = 1508 \quad \Sigma x ^ { 2 } = 51825$$

1. Calculate unbiased estimates of the population mean and variance.
2. Each student's mark is scaled using the formula \(y = 1.5 x + 10\). Find estimates of the population mean and variance of the scaled marks, \(y\).

1 The mass, in kilograms, of a block of cheese sold in a supermarket is denoted by the random variable \(M\). The masses of a random sample of 40 blocks are summarised as follows. $$n = 40 \quad \Sigma m = 20.50 \quad \Sigma m ^ { 2 } = 10.7280$$

1. Calculate unbiased estimates of the population mean and variance of \(M\).
2. The price, \(
   ) P\(, of a block of cheese of mass \)M \mathrm {~kg}\( is found using the formula \)P = 11 M + 0.50\(. Find estimates of the population mean and variance of \)P$.

1 The amounts of electricity, \(x \mathrm { kWh }\) (kilowatt hours), used by 40 households in a three-month period are summarised as follows. $$n = 40 \quad \sum x = 59972 \quad \sum x ^ { 2 } = 96767028$$

1. Calculate the mean and standard deviation of \(x\).
2. The formula \(y = 0.163 x + 14.5\) gives the cost in pounds of the electricity used by each household. Use your answers to part (i) to deduce the mean and standard deviation of the costs of the electricity used by these 40 households.

1 The hourly wages, \(\pounds x\), of a random sample of 60 employees working for a company are summarised as follows. $$n = 60 \quad \sum x = 759.00 \quad \sum x ^ { 2 } = 11736.59$$

1. Calculate the mean and standard deviation of \(x\).
2. The workers are offered a wage increase of \(2 \%\). Use your answers to part (i) to deduce the new mean and standard deviation of the hourly wages after this increase.
3. As an alternative the workers are offered a wage increase of 25 p per hour. Write down the new mean and standard deviation of the hourly wages after this 25p increase.

2 Dwayne is a car salesman. The numbers of cars, \(x\), sold by Dwayne each month during the year 2008 are summarised by $$n = 12 , \quad \Sigma x = 126 , \quad \Sigma x ^ { 2 } = 1582 .$$

1. Calculate the mean and standard deviation of the monthly numbers of cars sold.
2. Dwayne earns \(\pounds 500\) each month plus \(\pounds 100\) commission for each car sold. Show that the mean of Dwayne's monthly earnings is \(\pounds 1550\). Find the standard deviation of Dwayne's monthly earnings.
3. Marlene is a car saleswoman and is paid in the same way as Dwayne. During 2008 her monthly earnings have mean \(\pounds 1625\) and standard deviation \(\pounds 280\). Briefly compare the monthly numbers of cars sold by Marlene and Dwayne during 2008.

2 The hourly wages, \(\pounds x\), of a random sample of 60 employees working for a company are summarised as follows. $$n = 60 \quad \sum x = 759.00 \quad \sum x ^ { 2 } = 11736.59$$

1. Calculate the mean and standard deviation of \(x\).
2. The workers are offered a wage increase of \(2 \%\). Use your answers to part (i) to deduce the new mean and standard deviation of the hourly wages after this increase.
3. As an alternative the workers are offered a wage increase of 25 p per hour. Write down the new mean and standard deviation of the hourly wages after this 25p increase.

3 Dwayne is a car salesman. The numbers of cars, \(x\), sold by Dwayne each month during the year 2008 are summarised by $$n = 12 , \quad \Sigma x = 126 , \quad \Sigma x ^ { 2 } = 1582 .$$

1. Calculate the mean and standard deviation of the monthly numbers of cars sold.
2. Dwayne earns \(\pounds 500\) each month plus \(\pounds 100\) commission for each car sold. Show that the mean of Dwayne's monthly earnings is \(\pounds 1550\). Find the standard deviation of Dwayne's monthly earnings.
3. Marlene is a car saleswoman and is paid in the same way as Dwayne. During 2008 her monthly earnings have mean \(\pounds 1625\) and standard deviation \(\pounds 280\). Briefly compare the monthly numbers of cars sold by Marlene and Dwayne during 2008.

5 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.

1. Calculate the mean and standard deviation of the numbers of loaves bought per person.
2. Each loaf costs \(\pounds 1.04\). Calculate the mean and standard deviation of the amount spent on loaves per person.

1 The weights, \(x\) grams, of 100 potatoes are summarised as follows. $$n = 100 \quad \sum x = 24940 \quad \sum x ^ { 2 } = 6240780$$

1. Calculate the mean and standard deviation of \(x\).
2. The weights, \(y\) grams, of the potatoes after they have been peeled are given by the formula \(y = 0.9 x - 15\). Deduce the mean and standard deviation of the weights of the potatoes after they have been peeled.

1. The heights, \(x\) metres, of 40 children were recorded by a teacher. The results are summarised as follows

$$\sum x = 58 \quad \sum x ^ { 2 } = 84.829$$

1. Find the mean and the variance of the heights of these 40 children. The teacher decided that these statistics would be more useful in centimetres.
2. Find
   1. the mean of these heights in centimetres,
   2. the standard deviation of these heights in centimetres. Two more children join the group. Their heights are 130 cm and 160 cm .
3. 1. State, giving a reason, the mean height of the 42 children.
   2. Without recalculating the standard deviation, state, giving a reason, whether the standard deviation of the heights of the 42 children will be greater than, less than or the same as the standard deviation of the heights of the group of 40 children.