The study team enrolled 100 women aged 20-39 whom had identified as having on average less than 3 hours sleep per night. In a double-blinded randomised experiment, 50 of the participants were given Betasnooze to self-administer over a 3-month period, while the other 50 participants were given a placebo. The participants were required to report whether their sleep had improved after taking the full course of the drug. They were also asked whether they had been taking the oral contraceptive pill over the study period. The following results were generated from the trial: Table 3.1: Trial data from the double-blinded randomised controlled trail of Betasnooze (N=100). Trial Arm Placebo Betasnooze Oral contraceptive pill (OCP) OCP No OCP OCP No OCP Total participants 27 23 28 22 Participants with improved sleep 9 7 13 16 Assume that the population probability of improved sleep in each of the four groups (in the order presented) is given by 0,0, a, n. a) Write out the likelihood L(0,0, a, n) and the log-likelihood e(0,0,a,n), before finding the maximum likelihood estimate (MLE) for each of the four parameters. [2 marks] The study team enrolled 100 women aged 20-39 whom had identified as having on average less than 3 hours sleep per night. In a double-blinded randomised experiment, 50 of the participants were given Betasnooze to self-administer over a 3-month period, while the other 50 participants were given a placebo. The participants were required to report whether their sleep had improved after taking the full course of the drug. They were also asked whether they had been taking the oral contraceptive pill over the study period. The following results were generated from the trial: Table 3.1: Trial data from the double-blinded randomised controlled trail of Betasnooze (N=100). Trial Arm Placebo Betasnooze Oral contraceptive pill (OCP) OCP No OCP OCP No OCP Total participants 27 23 28 22 Participants with improved sleep 9 7 13 16 Assume that the population probability of improved sleep in each of the four groups (in the order presented) is given by 0,0, a, n. a) Write out the likelihood L(0,0, a, n) and the log-likelihood e(0,0,a,n), before finding the maximum likelihood estimate (MLE) for each of the four parameters. [2 marks]