Question 3 - A-Level Maths

In a game, a coin is spun 5 times and the number of heads obtained is recorded. Tao suggests playing the game 20 times and carrying out a chi-squared test to investigate whether the coin might be biased.
1. Explain why playing the game only 20 times may cause problems when carrying out the test.
Chris decides to play the game 500 times. The results are as follows
Number of heads 0 1 2 3 4 5
Observed frequency 2 27 93 181 146 51
Chris decides to test whether or not the data can be modelled by a binomial distribution, with the probability of a head on each spin being 0.6 She calculates the expected frequencies, to 2 decimal places, as follows
Number of heads 0 1 2 3 4 5
Expected frequency 5.12 38.40 115.20 172.80 129.60 38.88
State the number of degrees of freedom in Chris' test, giving a reason for your answer.
Carry out the test at the \(5 \%\) level of significance. You should state your hypotheses, test statistic, critical value and conclusion clearly.
Showing your working, find an alternative model which would better fit Chris’ data.

Number of heads	0	1	2	3	4	5
Expected frequency	5.12	38.40	115.20	172.80	129.60	38.88

Number of heads	0	1	2	3	4	5
Observed frequency	2	27	93	181	146	51