AQA A-Level Further Maths 2027 Predicted Practice Paper — Paper 3 Statistics

Answer ALL questions. Write your answers in the spaces provided. You must write down all the stages in your working.

Section A

Short and medium-length questions. Answer ALL questions.

• 1

  Question SECTION-A1 (6 marks)

  A quality inspector models the mass, X grams, of the sugar dispensed into a jar by a filling machine as a continuous random variable with probability density function f(x) = k*x for 0 <= x <= 4, f(x) = 0 otherwise, where k is a constant. (a) Show that k = 1/8. (2 marks) (b) Find E(X). (2 marks) (c) Find P(X > 3). (2 marks)

  (Total for Question SECTION-A1 is 6 marks)
• 2

  Question SECTION-A2 (7 marks)

  A machine that packs flour is supposed to dispense a mean mass of 1000 g. The masses are normally distributed with known standard deviation sigma = 6.8 g. A random sample of 40 bags is taken. To simplify the arithmetic, the inspector codes each mass as Y = (mass in grams) - 950, and finds the coded sample mean to be ybar = 52.4 (so the true sample mean mass is 1002.4 g). Note that Y also has standard deviation 6.8 g. (a) Calculate a 95% confidence interval for the population mean, giving your endpoints on the coded scale Y and also as true masses in grams. (4 marks) (b) State one assumption about the distribution of the sample mean that is needed for this interval, and explain why it holds exactly here. (1 mark) (c) Explain briefly what is meant by a '95% confidence interval'. (2 marks)

  (Total for Question SECTION-A2 is 7 marks)
• 3

  Question SECTION-A3 (8 marks)

  Two filling machines, A and B, each fill 500 ml bottles. The volumes are normally distributed with known standard deviations sigma_A = 3.2 ml and sigma_B = 3.8 ml. A random sample of 50 bottles from A has mean 248.6 ml (measured as 'ml above 250') and a random sample of 45 bottles from B has mean 246.9 ml on the same scale. A supervisor wishes to test, at the 5% significance level, whether the two machines dispense different mean volumes. (a) State suitable null and alternative hypotheses. (1 mark) (b) Carry out the test, showing your test statistic and conclusion in context. (7 marks) [Total marks for this question: 8]

  (Total for Question SECTION-A3 is 8 marks)
• 4

  Question SECTION-A4 (7 marks)

  A researcher records the number of hours of revision, x, and the exam score, y, for 15 randomly selected students and calculates the product-moment correlation coefficient to be r = 0.62. (a) Test, at the 5% significance level, whether there is positive correlation between revision hours and exam score. Use the critical value 0.4409. (4 marks) (b) A student claims that increasing revision hours causes higher exam scores. State, with a reason, whether the test in part (a) supports this claim. (2 marks) (c) Give one reason why a product-moment correlation coefficient might be an inappropriate summary for a particular bivariate data set. (1 mark) [Total marks for this question: 7]

  (Total for Question SECTION-A4 is 7 marks)
• 5

  Question SECTION-A5 (7 marks)

  In a fairground game a player pays to spin a wheel once. The random variable W represents the player's net winnings, in GBP, and has the following probability distribution: w: 5 2 -3 P(W=w): 0.2 0.3 0.5 (a) Find E(W) and interpret its sign for the player. (3 marks) (b) Find Var(W). (3 marks) (c) The organiser runs the game 400 times in a day, with spins independent. Using your answer to (a), estimate the organiser's expected total profit for the day. (1 mark) [Total marks for this question: 7]

  (Total for Question SECTION-A5 is 7 marks)
• 6

  Question SECTION-A6 (8 marks)

  A large batch of electronic components has a defect rate of p = 0.35. A random sample of 200 components is inspected and X is the number found to be defective. (a) State the exact distribution of X and give its mean and standard deviation. (2 marks) (b) Using a suitable approximation with a continuity correction, estimate P(X >= 80). (4 marks) (c) Give one reason why the approximation used in part (b) is justified here. (2 marks) [Total marks for this question: 8]

  (Total for Question SECTION-A6 is 8 marks)
• 7

  Question SECTION-A7 (6 marks)

  A geneticist crosses plants and expects offspring to fall into four phenotype categories in the ratio 3 : 3 : 2 : 2. In a sample of 160 offspring the observed frequencies are 42, 55, 38 and 25 respectively. (a) Show that the expected frequency for the first category is 48. (1 mark) (b) Carry out a chi-squared goodness-of-fit test at the 5% significance level to test whether the data are consistent with the expected ratio. State your hypotheses, test statistic, degrees of freedom and conclusion. (5 marks) [Total marks for this question: 6]

  (Total for Question SECTION-A7 is 6 marks)
• 8

  Question SECTION-A8 (8 marks)

  A market researcher wishes to test whether preference for one of two supermarket brands is associated with age group. A random sample of 200 shoppers gives the following contingency table of counts: Under 35 35-54 55+ Brand P 40 30 30 Brand Q 20 45 35 (a) Stating your hypotheses, calculate the expected frequency for shoppers aged Under 35 who prefer Brand P. (3 marks) (b) Given that the chi-squared test statistic is 10.05, test at the 5% significance level whether brand preference is independent of age group. State the degrees of freedom, the critical value and your conclusion in context. (5 marks) [Total marks for this question: 8]

  (Total for Question SECTION-A8 is 8 marks)

Section B

Extended multi-part questions. Answer ALL questions.

• 1

  Question SECTION-B1 (14 marks)

  A helpline models the number of calls it receives as a Poisson process. During quiet periods the mean rate is 4.2 calls per hour, and calls are received independently. (a) Find the probability that in a randomly chosen quiet hour the helpline receives (i) no calls, and (ii) at least 3 calls. (4 marks) (b) Find the probability that the helpline receives at least 3 calls in exactly two of four independent quiet hours. (4 marks) (c) The manager suspects the rate has increased. Over a randomly chosen 8-hour quiet shift, 45 calls are recorded. Using a suitable normal approximation with a continuity correction, test at the 5% significance level whether the mean call rate has increased. State your hypotheses, test statistic and conclusion in context. (6 marks) [Total marks for this question: 14]

  (Total for Question SECTION-B1 is 14 marks)
• 2

  Question SECTION-B2 (15 marks)

  A biologist studies the relationship between water temperature x (degrees C) and the growth rate y (mm per day) of an algae culture. For n = 12 observations the following summary statistics are obtained: mean of x = 15.0, mean of y = 22.0, S_xx = 210.0, S_xy = 168.0, S_yy = 150.0. (a) Calculate the equation of the least-squares regression line of y on x, giving it in the form y = a + b*x. (5 marks) (b) Calculate the product-moment correlation coefficient r, and hence state the proportion of the variation in y explained by the linear regression. (4 marks) (c) Use your regression line to estimate the growth rate when the water temperature is 18 degrees C, and give one reason why an estimate at 30 degrees C would be much less reliable. (3 marks) (d) One further data point is found to have a large positive residual. Explain what a residual is and what a large positive residual tells you about this point. (3 marks) [Total marks for this question: 15]

  (Total for Question SECTION-B2 is 15 marks)
• 3

  Question SECTION-B3 (14 marks)

  A pharmaceutical trial compares a new painkiller against a placebo. Each of 190 patients reports whether they experienced 'significant relief', and the results are cross-classified by treatment: Relief No relief New drug 64 26 Placebo 36 64 (a) Carry out a chi-squared test for association at the 1% significance level. State your hypotheses, calculate the expected frequencies, compute the test statistic (you may use the uncorrected form), state the degrees of freedom and critical value, and give your conclusion in context. (9 marks) (b) Explain briefly why a continuity (Yates') correction is sometimes applied to a 2x2 chi-squared test, and state its qualitative effect on the test statistic. (3 marks) (c) State one limitation of concluding a causal effect of the drug from this test alone. (2 marks) [Total marks for this question: 14]

  (Total for Question SECTION-B3 is 14 marks)