Q Scheme Marks AOs Pearson Progression Step and Progress descriptor 1a Two from: Each bolt is either faulty or not faulty. The probability of a bolt being faulty (or not) may be assumed constant. Whether one bolt is faulty (or not) may be assumed to be independent (or does not affect the probability of) whether another bolt is faulty (or not). There is a fixed number (50) of bolts. A random sample. B2 1.2 1.2 5th Understand the binomial distribution (and its notation) and its use as a model. (2) 1b Let X represent the number of faulty X~B(50, 0.25) P(X ⩽ 6) = 0.0194 P(X ⩽ 7) = 0.0453 P(X ⩾ 19) = 0.0287 P(X ⩾ 20) = 0.0139 M1 M1dep 3.4 1.1b 5th Find critical values and critical regions for a binomial distribution. Critical Region is X ⩽ 6 X ⩾ 20 A2 1.1b 1.1b (4) (6 marks) Notes 1a Each comment must be in context for its mark. Q Scheme Marks AOs Pearson Progression Step and Progress descriptor 2a The set of values of the test statistic for which the null hypothesis is rejected in a hypothesis test. B2 1.2 1.2 5th Understand the language of hypothesis testing.
3c Not in critical region therefore insufficient evidence to reject H0. B1 2.2b 6th Interpret the results of a binomial distribution test in context. There is insufficient evidence at the 5% level to suggest that the value of p is not 0.3. B1 3.2a (2) (6 marks) Notes 3c Conclusion must contain context and non-assertive for first B1. Q Scheme Marks AOs Pearson Progression Step and Progress descriptor 4a X~B(28, 0.37) M1 3.4 5th Find critical values and critical regions for a binomial distribution P(X ⩾ 15) = 1 – 0.9454 = 0.0546 and P(X ⩾ 16) = 1 – 0.9762 = 0.0238 M1dep 1.1b Critical region is X ⩾ 16 A1 1.1b (3) 4b In critical region therefore sufficient evidence to reject H0 B1 2.2b 6th Interpret the results of a binomial distribution test in context. There is sufficient evidence at the 5% level to suggest that the value of p is bigger than 0.37. B1 3.2a (2) (5 Notes 4a First M1 for correct distribution seen or implied. Second M1 (dependent on first) for evidence that correct probabilities for either critical value examined. 4b Conclusion must contain context and non-assertive for first B1.
Q Scheme Marks AOs Pearson Progression Step and Progress descriptor 6a Let X represent the number of bowls with minor defects (seen or implied). XB(25, 0.2) may be implied P(X ⩽ l) = 0.0274 P(X = 0) = 0.0038 P(X ⩽ 8) = 0.9532  P(X ⩾ 9) = 0.0468 P(X ⩽ 9) = 0.9827  P(X ⩾ 10) = 0.0173 Critical region is X = 0 X ⩾ 10 M1 M1dep A1 M1 A2 3.4 1.1b 1.1b 1.1b 1.1b 1.1b 5th Find critical values and critical regions for a binomial distribution. (6) 6b Significance level = 0.0038 + 0.0173 = 0.0211 or 2.11% B1 1.2 6th Calculate actual significance levels for a binomial distribution test. (1) 6c H0: p = 0.2; H1: p < 0.2 B1 2.5 5th Carry out 1-tail tests for the binomial distribution. Let Y represent number of with minor defects (Under H0) Y~B(20, 0.2) (may be implied) M1 3.4 Either P(Y ⩽ 2) = 0.2061 0.2061 > 0.1 (or 10%) Insufficient evidence at the 10% to suggest that the proportion of defective bowls has decreased. Or P(Y ⩽ 2) = 0.2061 P(Y ⩽ 1) = 0.0692 so critical region is Y ⩽ 1 Insufficient evidence at the 10% level to suggest that the proportion of defective bowls has decreased. 0.2061 > 0.10 or 0.7939 < 0.9 B1 M1 A1 B1 M1 A1 1.1b 1.1b 3.2b 1.1b 1.1b 3.2a (5) (12