1.1. The table below shows the cumulative probabilities for different values that $X\textasciitilde{}B(10,0.5)$ can take:

Kieran collects coins and suspects that one of them is biased.  To test his suspicion Kieran flips the coin 10 times and records the number of times, $T$, that it lands on tails.

Stating your hypotheses clearly, find the critical regions for the test using a 10% level of significance.

1.2. Calculate the probability of incorrectly rejecting the null hypothesis.

1.3. The coin lands on heads on each of the 10 flips.  Kieran claims that the coin is definitely biased.

Comment on the validity of Kieran’s claim.

1.4. Describe one adjustment Kieran could make to his test to give a more reliable conclusion.