6 A machine is used to toss a coin repeatedly. Rosa believes that the outcome of each toss made by the machine is not independent of the previous toss. Rosa gets the machine to toss a coin 6 times and record the number of heads, \(X\), obtained. After recording the number of heads obtained, Rosa resets the machine and gets it to toss the coin 6 more times. Rosa again records the number of heads obtained and she repeats this procedure until she has recorded 88 independent values of \(X\).
- The sample mean and sample variance of \(X\) are 3.35 and 3.392 respectively.
Explain what these results suggest about the validity of a binomial model \(\mathrm { B } ( 6 , p )\) for the data.
Rosa uses a computer spreadsheet to work out the probabilities for a more sophisticated model in which the outcome of each toss is dependent on the outcome of the previous toss. Her model suggests that the probabilities \(\mathrm { P } ( X = x )\), for \(x = 0,1,2,3,4,5,6\), are approximately in the ratio \(5 : 6 : 7 : 8 : 7 : 6 : 5\). She carries out a \(\chi ^ { 2 }\) test to investigate whether this model is a good fit for the data.
The following table shows the full results of the experiments, together with some of the calculations needed for the test.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | Total |
| Observed frequency | 7 | 10 | 16 | 15 | 15 | 11 | 14 | 88 |
| Expected frequency | | | | | | | | |
| Contribution to \(\chi ^ { 2 }\) statistic | 0.9 | 0.3333 | 0.2857 | 0.0625 | 0.0714 | | | |
- In the Printed Answer Booklet, complete the table.
- Carry out the test, using a 10\% significance level.
- Rosa says that the results definitely show that one of the two proposed models is correct.
Comment on this statement.