Mathematics

1. 1.

   X follows a binomial distribution with n = 10 and p = 0.3. Calculate P(X >= 3).

   [3 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. 'At least 3' is the complement of 'at most 2', which needs far fewer terms.
   2. Use the cumulative binomial function on the calculator.

   Marking points

   • Writes P(X >= 3) = 1 - P(X <= 2).
   • Finds P(X <= 2) = 0.3828 from tables or a calculator.
   • P(X >= 3) = 0.6172.

   Examiner tip: P(X >= 3) = 1 - P(X <= 2), not 1 - P(X <= 3). Check the inequality direction.

2. 2.

   The heights of adult men are modelled by a normal distribution with mean 170 cm and standard deviation 8 cm. Calculate the probability that a randomly chosen man is taller than 182 cm.

   [3 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Convert to the standard normal using z = (x - mean)/standard deviation.
   2. The question asks for the upper tail, so subtract the table value from 1.

   Marking points

   • Standardises: z = (182 - 170)/8 = 1.5.
   • Uses P(Z > 1.5) = 1 - P(Z < 1.5).
   • The probability is 0.0668.

   Examiner tip: Sketch the curve to decide whether you need the area to the left or right of z.

3. 3.

   For events A and B, P(A) = 0.6, P(B) = 0.5 and P(A and B) = 0.35. Find P(A | B), determine whether A and B are independent, and find P(A or B).

   [4 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Conditional probability divides the joint probability by the probability of the condition.
   2. The addition law subtracts the overlap once to avoid double counting.

   Marking points

   • P(A | B) = P(A and B)/P(B) = 0.35/0.5 = 0.7.
   • Independence needs P(A)P(B) = P(A and B).
   • 0.6 * 0.5 = 0.3, which is not 0.35, so they are not independent.
   • P(A or B) = 0.6 + 0.5 - 0.35 = 0.75.

   Examiner tip: To test independence compare P(A)P(B) with P(A and B). Do not assume independence.

4. 4.

   A coin is tossed 20 times and lands heads 15 times. Test, at the 5% significance level, whether the coin is biased towards heads. State your hypotheses and conclusion.

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The question says 'biased towards heads', so the alternative is one-tailed in the upper direction.
   2. Compare the probability of a result as extreme as 15 with the 5% level.

   Marking points

   • H0: p = 0.5 and H1: p > 0.5 (one-tailed).
   • Under H0, X ~ B(20, 0.5) and the p-value is P(X >= 15).
   • P(X >= 15) = 0.0207.
   • 0.0207 < 0.05, so reject H0: there is evidence that the coin is biased towards heads.

   Examiner tip: Use P(X >= 15), not P(X = 15). Conclude in context, not just 'reject H0'.

5. 5.

   A machine fills bottles with mean volume 250 ml and known standard deviation 6 ml. A sample of 36 bottles has mean 247.8 ml. Test at the 5% level whether the mean volume has decreased.

   [5 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The standard deviation of a sample mean is sigma/sqrt(n), not sigma.
   2. A one-tailed test at 5% uses the critical value -1.645; compare the standardised result with it.

   Marking points

   • H0: mu = 250 and H1: mu < 250 (one-tailed).
   • The sample mean is distributed N(250, 6^2/36), standard error 1.
   • Test statistic z = (247.8 - 250)/1 = -2.2.
   • Compares with the critical value -1.645 (or p-value 0.0139 < 0.05).
   • Rejects H0: there is evidence that the mean volume has decreased.

   Examiner tip: Use the standard error sigma/sqrt(n) for a sample mean. Write the conclusion in the context of the machine.

6. 6.

   For 8 pairs of data (x, y) with x between 2 and 9: sum x = 40, sum y = 96, Sxx = 60, Sxy = 84, Syy = 120. Find the regression line of y on x, predict y when x = 7, find the product moment correlation coefficient, and interpret the gradient.

   [6 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Compute the gradient from the summary statistics, then the intercept from the means (the line passes through (x-bar, y-bar)).
   2. Predicting within the range of the data is interpolation and reliable; outside the range is extrapolation and risky.

   Marking points

   • Gradient b = Sxy/Sxx = 84/60 = 1.4.
   • Means: x-bar = 5 and y-bar = 12, so a = 12 - 1.4 * 5 = 5.
   • The regression line is y = 5 + 1.4x.
   • At x = 7: y = 14.8 (reliable, as 7 is within the data range 2 to 9).
   • r = 84/sqrt(60 * 120) = 0.990, a very strong positive correlation.
   • Interpretation: each extra unit of x is associated with an increase of 1.4 units in y.

   Examiner tip: Interpret the gradient in context and mention interpolation. Correlation does not by itself prove cause.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.