Hints offered by N Hopley

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Paper 1

Question 1

1a) Hint 1: recognise this is a Poisson distribution

1a) Hint 2: note that P(X < 3) = P(X ≤ 2)

1b) Hint 3: note that the time period has changed from 1 day to 7 days

1b) Hint 4: create a new Poisson random variable with a new parameter

Question 2

2a) Hint 1: note that the word 'given' means conditional probability

2a) Hint 2: given that they are colour blind, just use the second row of the table to obtain the numbers required

2b) Hint 3: know that if A and B are independent events, then P(A | B) = P(A)

2b) Hint 4: work out P(left handed) by using the first column of the table

2b) Hint 5: write words to explain the logic behind your conclusion from comparing probabilities

2c) Hint 6: perform a standard chi-squared test of association

2c) Hint 7: be sure to comment on the conclusion in terms of the study i.e. in terms of sex and colour blindness and for which it's more likely

Question 3

3a) Hint 1: note that the sample has not used random numbers, so it is a non-random sampling method

3a) Hint 2: determine whether this means it is either convenience or quota sampling

3b) Hint 3: be sure to write as much detail as possible, including the use of random numbers

Question 4

Hint 1: note that this is not a chi-squared test, as we are not testing for an association

Hint 2: note that Team A's and Team B's results can be combined together

Hint 3: recognise that this is a proportion test over a total of 64 games

Hint 4: note that the test statistic is the proportion of wins when the team scored first, which is 37 out of 64

Question 5

5a) Hint 1: work out the mean and variance of the continuous uniform distribution, using the formulae from the Data Booklet

5a) Hint 2: state the parameters of the approximate normal distribution, for the sample mean that uses a sample of size 25

5a) Hint 3: calculate the probability that the sample mean is greater than 16.7

5a) Hint 4: note that continuity correction is not needed here as the uniform distribution was continuous, and not discrete

5b) Hint 5: know that the CLT can be justified when the sample size is greater than 20

Question 6

6a) Hint 1: recognise that 'species' is categorical data and 'mean energy' is continuous data

6a) Hint 2: know the types of charts which can show both these types of data at the same time

6b) Hint 3: know that any confidence interval from a sample estimate requires the population to be normally distributed

6b) Hint 4: use the sample standard deviation formula on page 4 of the Data Booklet to calculate the estimate for the population standard deviation, from the sample

6b) Hint 5: recognise that we have approximated the population standard deviation, so we need to use the t distribution

6b) Hint 6: know that the t distribution will have 10-1 = 9 degrees of freedom

6b) Hint 7: proceed with standard method of calculating a confidence interval

6c) Hint 8: look at ALL of the mean energy values to see which ones lie inside the confidence interval. There should be more than one.

6d) Hint 9: re-calculate the confidence interval and again scan the full list of all mean energy values

Question 7

7a) Hint 1: note that the sample size is 6, not 8

7a) Hint 2: calculate the mean of the eight Daily Mean values

7a) Hint 3: calculate the mean of the eight Daily Range values

7a) Hint 4: use the given formula to determine the value of σ-hat

7a) Hint 5: use the given expression to evaluate the 3-sigma limits

7b) Hint 6: carefully substitute the correct values for the correct terms and solve for d

Question 8

8a) Hint 1: know what a positive linear correlation looks like

8b) Hint 2: use the formula at the top of page 5 of the Data Booklet to work out r, then work out r²

8b) Hint 3: know that the coefficient of determination measures the percent of variability explained by the linear regression model

8c) Hint 4: recognise that this is a standard hypothesis test on ρ

8c) Hint 5: use the test statistic of r, from part (b)

8c) Hint 6: know that any test using the t distribution requires the assumption that the population from which the sample is taken, is normally distributed

8d) Hint 7: know that a residual plot is used to check the fit of a linear regression model

8d) Hint 8: know that the residual plot should show the mean residual to be 0 and the variance to be constant

Question 9

Hint 1: know that a single sample Wilcoxon test assumes that the population is distributed symmetrically

Hint 2: perform a standard Wilcoxon test on a single sample

Hint 3: look out for any values that equal the median value of 65, as they will be excluded from the analysis

Hint 4: realise that we have a sample size > 20 meaning that a normal approximation is required

Hint 5: remember to use continuity correction

Question 10

10a) Hint 1: recognise that this is a z-test as we have a population with a normal distribution and known variance

10b) Hint 2: process your workings for 5% and then again for 1%

10b) Hint 3: write words to explain what happens for both above and below each of the calculated values

10c) Hint 4: know that if we had to estimate the population variance, then a t distribution would have to be used

10c) Hint 5: know that the t distribution will have 25-1 = 24 degrees of freedom

Question 11

11a) Hint 1: draw a tree diagram for Instructor A, with the first branches splitting into 'pass on 1st attempt' and 'not pass on 1st attempt'

11a) Hint 2: label the second branches coming off from 'not pass on 1st attempt' to be 'pass on 2nd attempt' and 'not pass on 2nd attempt'

11b)i) Hint 3: draw a tree diagram for Instructor B, with the same design as the previous tree diagram

11b)i) Hint 4: label your tree diagram branches with all the probabilities that you know

11b)ii) Hint 5: recognise that this is a conditional probability question that your second tree diagram can help you answer

11c) Hint 6: recognise that this is a conditional probability question where the condition is now on 'failing after 2 attempts'

Question 12

12a) Hint 1: recognise that this is a Binomial trials situation

12a) Hint 2: determine that the number of 1's on show is distributed as B(3, 1/8)

12a) Hint 3: round the exact fractional answers to 4 decimal places

12a) Hint 4: calculate E(X) and SD(X) to match the values provided in the question

12b)i) Hint 5: recognise that we require the sum of 60 values of X and 45 values of Y, but this is __not__ the same as 60X+45Y

12b)i) Hint 6: know that the variance formula for a linear combination of random variables requires all of the random variables to be independent

12b)ii) Hint 7: know such facts that we'd expect about 95% of outcomes to lie within 2 standard deviations of the mean

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