A coin was tossed 200 times and 115 of these tosses resulted in ‘heads’. Use a two-tailed test with significance level 1 % to investigate whether or not the coin is biased.

The number of’ ‘heads’ X is B(200, p)     (M1)

\({{\text{H}}_0}:p = 0.5;{\text{ }}{{\text{H}}_1}:p \ne 0.5\)     A1A1

Note: Award A1A0 for the statement “ \({{\text{H}}_0}:\) coin is fair; \({{\text{H}}_1}:\) coin is biased”.

EITHER

\({\text{P}}(\left. {X \geqslant 115} \right|{{\text{H}}_0}) = 0.0200\)     (M1)(A1)

p-value = 0.0400     A1

This is greater than 0.01.     R1

There is insufficient evidence to conclude that the coin is biased (or the coin is not biased).     R1

OR

(Using a proportion test on a GDC) p-value = 0.0339     N3

This is greater than 0.01.     R1

There is insufficient evidence to conclude that the coin is biased (or the coin is not biased).     R1

OR

Under \({{\text{H}}_0}X\) is approximately N(100, 50)     (M1)

\(z = \frac{{115 - 100}}{{\sqrt {50} }} = 2.12\)     (M1)A1

(Accept 2.05 with continuity correction)

This is less than 2.58     R1

There is insufficient evidence to conclude that the coin is biased (or the coin is not biased).     R1

OR

99 % confidence limits for p are \(\frac{{115}}{{200}} \pm 2.576\sqrt {\frac{{115}}{{200}} \times \frac{{85}}{{200}} \times \frac{1}{{200}}} \)     (M1)A1

giving [0.485, 0.665]     A1

This interval contains 0.5     R1

There is insufficient evidence to conclude that the coin is biased (or the coin is not biased).     R1

[8 marks]

This question was well answered in general with several correct methods seen. The most popular method was to use a GDC to carry out a proportion test which is equivalent to using a normal approximation. Relatively few candidates calculated an exact p-value using the binomial distribution. Candidates who found a 95% confidence interval for p, the probability of obtaining a head, and noted that this contained 0.5 were given full credit.