Part 1

First we need to calculate the mean. Since each item has an equal probability, we can apply a simple arithmetic average.

Part 2

The variance can be calculated using the following formula:

However, since all the probabilities are equal to 1/4, this can be rewritten as:

Substituting our values gives:

Therefore the standard deviation is just:

Frequency To Probabilities

We can convert each frequency into a probability by dividing it by the total number of runs.

Therefore, the probabilities can be expressed as:

Finding The Expected Value:

Therefore, her average run is around 2.68km

Finding The Variance:

The variance can be calculated as:

Therefore, we must find :

The variance can then be written as:

Finding the Standard Deviation

The standard deviation can be found by square rooting the variance:

Part 1

The expected value of can be found using the following formula:

Part 2

If we want to find using the supplied formula, we must create a row for in our probability distribution table. This can be seen below:

We can therefore express the variance as:

Part 3

If we want to find using the supplied formula, we must create a row for in our probability distribution table. This can be seen below:

We can now use this table to find :

We can now use the supplied formula to find our variance:

Note: It is important to realise that the methods in part 2 and 3 both return the same result.

Part 1

Frequencies can be converted into probabilities by dividing each one by the sum of frequencies ( number of people in the survey ).

We can now convert the data into a probability distribution table as seen below:

Part 2

The average shoe size can be calculated by finding the expected value of the probability distribution table:

Note: You can utilise the expected value feature in your calculator to help compute this. Try google search “Calculate expected value your calculator model”.

Part 3

We must find to calculate the variance. This is shown below:

We can then calculate the variance as:

Therefore the standard deviation is simply:

Note: You can utilise the variance and standard deviation feature in your calculator to compute these values directly. Try google search “Calculate standard deviation your calculator model”.

Overall Approach

We can solve this question by utilising the following variance formula:

Therefore, we must find and . Once we find these values, we can substitute them back into the variance formula and solve for k.

Finding The Expected Value:

Finding The Expected Value Of The Square:

Expressing The Variance:

We know that the variance of is 4.2. Substituting this value gives:

Solving The Quadratic

We now have a quadratic which we can solve using the quadratic formula.

Note: k can have either value and still create a variance of 4.2. Therefore both values of k are valid.

Part 1

We can create a probability distribution that maps each possible mark to a probability . Note: there are 20 students in the class.

Part 2

Finding the expected value of by hand is extremely tedious for this distribution. It will be much easier to use the statistical analysis tools built into most calculators. If you don’t know how to do this, try google search “Calculate expected value your calculator model”.

Part 3

Finding the standard deviation of by hand is extremely tedious for this distribution. It will be much easier to use the statistical analysis tools built into most calculators. If you don’t know how to do this, try google search “Calculate standard deviation your calculator model”.

If you do not know how to use the statistics mode in your calculator, try google search “Calculate expected value your calculator model” and “Calculate standard deviation your calculator model”

Part 1

We begin by writing the probability distribution table in our calculator and finding or depending on your calculator:

Part 2

Most calculators help you calculate . We can then find the variance by squaring this value: