Hints offered by N Hopley

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Paper 1

Question 1

1a)i) Hint 1: draw a tree diagram with survive/no survive first month as its first branches

1a)i) Hint 2: continue tree diagram with branches for reaching adulthood/not reaching adulthood

1a)i) Hint 3: continue tree diagram with branches for returning to wild/not returning to wild

1a)i) Hint 4: for required probability, it will be calculated along three branches of your tree diagram

1a)ii) Hint 5: for required probability, it will also be calculated along three branches of your tree diagram

1b) Hint 6: recognise that this is a conditional probability, conditional upon them not reaching adulthood

Question 2

2a) Hint 1: draw out a table for the five stated values of x, with a constant probability for all of them

2a) Hint 2: know that all the probabilities should sum to 1, if this is to be a valid random variable

2b)i) Hint 3: use standard formulae for E(X) and V(X)

2b)i) Hint 4: you need to work out E(X²) to then calculate V(X)

2b)ii) Hint 5: use your answer from part (b)(i) for the value of μ

Question 3

3a)i) Hint 1: bring your worldly experience of knowledge of how exam marks are - or are not - made public

3a)ii) Hint 2: for a random sample method, be sure to mention how random numbers are used

3a)ii) Hint 3: be sure to communicate as much detail as possible, being clear how your chosen method minimises the travelling distance

3b) Hint 4: recognise that you have two samples of non-paired data

3b) Hint 5: know that this is a t-test for a difference in population means

3b) Hint 6: know that this will involve pooling the samples to obtain the best estimate for the (equal) standard deviations

3b) Hint 7: make it clear what level of significance you have chosen for your test

3b) Hint 8: be sure to state your conclusion in terms of the context

Question 4

4a) Hint 1: standard process to calculate lower and upper fences: Q1 - 1.5×(Q3 - Q1) and Q3 + 1.5×(Q3 - Q1)

4b)i) Hint 2: recognise that the shoe/height data set is paired data

4b)ii) Hint 3: know what type of graph ought to be used to display paired data

4b)ii) Hint 4: know what sort of processes/tests could be used with a scatterplot of data

Question 5

5a)i) Hint 1: note that the sample size, n = 5

5a)i) Hint 2: state the mean and standard deviation of X̄ based upon the mean and standard deviation of X

5a)i) Hint 3: use a process similar to calculating 1σ, 2σ and 3σ limits, but use the number 6 instead

5a)ii) Hint 4: use the provided control chart to identify the first point that is higher than the 6σ limit line

5b)i) Hint 5: if the process is in control, then it should be equally likely to be either above or below the centre line

5b)i) Hint 6: recognise that this is a Binomial distribution with n = 9 and p = ½

5b)i) Hint 7: be aware that the 9 points could either be all above, or all below, the centre line

5b)ii) Hint 8: scrutinise the provided control chart to identify which point(s) meet the stated conditions

Question 6

Hint 1: recognise that we have a total of 260 random variables, all being added together

Hint 2: know that the formula for the variance of the total requires all 260 random variables to be independent of each other

Hint 3: you should not be squaring 130, as it's not V(130X) but rather V(X₁ + X₂ + … + X_{130})

Question 7

7a) Hint 1: know that a confidence interval is based upon the assumption that the population is normally distributed

7a) Hint 2: know that by estimating the population variance, we will therefore be using a t-distribution

7a) Hint 3: know that the t-distribution will have 17 degrees of freedom

7b)i) Hint 4: determine if the value 5.87 is within the confidence interval from part (a)

7b)i) Hint 5: communicate the meaning of the location of 5.87, relative to the confidence interval

7b)ii) Hint 6: consider whether there are any other possible influencing factors that might affect the result

Question 8

8a) Hint 1: know that 4 is the variance, so the standard deviation is not 4

8b) Hint 2: again, know that 4 is the variance, so the standard deviation is not 4

8c) Hint 3: use standard formulae to calculate E(Y) and V(Y) to allow you know SD(Y)

8c) Hint 4: draw a diagram of a uniform distribution to help visualise the interval required and the constant value of the probability density over that interval

Question 9

9a) Hint 1: consider the practical issues that might occur when dividing land into square grids and what may or may not be in each grid

9a) Hint 2: provide a practical suggestion, based upon common sense

9b) Hint 3: recognise that we do not know the type of distribution of X, but that we do know E(X) and V(X)

9b) Hint 4: notice that we have a sample size of 25, which is greater than 20

9b) Hint 5: recognise that this will require the Central Limit Theorem

9b) Hint 6: state that the distribution is approximately normal, and what its parameters are

9c) Hint 7: realise that we could assume the standard deviation here to be the same as that from part (b)

9c) Hint 8: recognise that we now have a single sample z-test of a mean

Question 10

10a) Hint 1: recognise that we have 20 repetitions of an action that had a probability of success to be 0.2

10b) Hint 2: from part (a) we know we have a binomial distribution, so this part requires you to first specify the new binomial distribution

10b) Hint 3: note that the 'appropriate approximation' means that you need to approximate a binomial with a normal

10b) Hint 4: as we are going from a discrete distribution to a continuous distribution, we need to use continuity correction

10c) Hint 5: the word 'association' is a clue to perform a chi-squared test, along with the obvious table of 2 rows and 2 columns

10c) Hint 6: be sure to state your conclusion in terms of the context

Question 11

11a) Hint 1: know that a random variable for a proportion stems from a random variable for a simple count, which is binomial

11a) Hint 2: therefore the binomial needs to be approximated by a normal distribution, to allow a division to occur to give a proportion

11a) Hint 3: know the expression for the standard deviation of this normal distribution, as it is not given in the formula booklet.

11b) Hint 4: recognise that this part requires you to 'work backwards' from a condition, to a sample size

11b) Hint 5: know that for a majority to support a claim, you need to be confident that more than 50% support it

11b) Hint 6: realise that we are seeking to ensure that the entire confidence interval is located above 50%,

11b) Hint 7: proceed with the lower value of the confidence interval is the proportion of 0.5 to obtain a lower bound for the sample size

Question 12

12a) Hint 1: know that a Mann Whitney test requires the assumption that the populations have the same shape and same spread, and that it is not paired data

12b)i) Hint 2: realise that with m = 20 and n = 20 we can use the data booklet tables, and that the situation is a 'symmetrical' one due to equal sample sizes

12b)i) Hint 3: perform a standard Mann Whitney test, using the provided rank sum value of 480

12b)ii) Hint 4: be sure to explain the different conclusion in terms of the context of the study

12b)iii) Hint 5: think of a practical step that could be done to improve the quality of the random sample, so that it is more representative of the population

Question 13

13a) Hint 1: make a comment about where most of the dots are, and how they are aligned

13a) Hint 2: make a second comment about the outlier dot and what it represents in terms of the data set

13b) Hint 3: use a standard process and formulae from the data booklet to obtain the value of r from the summary statistics provided

13b) Hint 4: square the value of r to obtain R² and know what this means in terms of a percentage of variation explained by the regression line

13c) Hint 5: note that the interval being asked for is not for a mean, but rather for an individual case

13c) Hint 6: proceed with calculating a prediction interval derived from the transformed x value of 2.5441 being substituted into the regression line equation

13c) Hint 7: note that the sample size, n = 13, which means that the t-distribution will have 11 degrees of freedom

13c) Hint 8: know that the resulting prediction interval is for the transformed number of species, and therefore its values need to be converted back to simply the number of species

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