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