5. A statistician believes a coin is biased and the probability, \(p\), of getting a head when the coin is tossed is less than 0.5 The statistician decides to test this by tossing the coin 10 times and recording the number, \(X\), of heads. He sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) and rejects the null hypothesis if \(x < 3\)

1. Find the size of the test.
2. Show that the power function of this test is $$( 1 - p ) ^ { 8 } \left( 36 p ^ { 2 } + 8 p + 1 \right)$$ Table 1 gives values, to 2 decimal places, of the power function for the statistician's test. \begin{table}[h] \end{table} Table 1
3. On the axes below draw the graph of the power function for the statistician's test.
4. Find the range of values of \(p\) for which the probability of accepting the coin as unbiased, when in fact it is biased, is less than or equal to 0.4
   \includegraphics[max width=\textwidth, alt={}, center]{1d84c9fc-be67-45be-b439-3111c48ff1cb-09_1143_1209_945_402}