Calculate Type II error probability

6 Jeevan thinks that a six-sided die is biased in favour of six. In order to test this, Jeevan throws the die 10 times. If the die shows a six on at least 4 throws out of 10 , she will conclude that she is correct.

1. State appropriate null and alternative hypotheses.
2. Calculate the probability of a Type I error.
3. Explain what is meant by a Type II error in this situation.
4. If the die is actually biased so that the probability of throwing a six is \(\frac { 1 } { 2 }\), calculate the probability of a Type II error.

4 It is not known whether a certain coin is fair or biased. In order to perform a hypothesis test, Raj tosses the coin 10 times and counts the number of heads obtained. The probability of obtaining a head on any throw is denoted by \(p\).

1. The null hypothesis is \(p = 0.5\). Find the acceptance region for the test, given that the probability of a Type I error is to be at most 0.1 .
2. Calculate the probability of a Type II error in this test if the actual value of \(p\) is 0.7 .

8 The random variable \(R\) has the distribution \(\mathrm { B } ( 14 , p )\). A test is carried out at the \(\alpha \%\) significance level of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\), against \(\mathrm { H } _ { 1 } : p > 0.25\).

1. Given that \(\alpha\) is as close to 5 as possible, find the probability of a Type II error when the true value of \(p\) is 0.4 .
2. State what happens to the probability of a Type II error as
   (a) \(p\) increases from 0.4,
   (b) \(\alpha\) increases, giving a reason.

4. A proportion \(p\) of letters sent by a company are incorrectly addressed and if \(p\) is thought to be greater than 0.05 then action is taken.
Using \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\), a manager from the company takes a random sample of 40 letters and rejects \(\mathrm { H } _ { 0 }\) if the number of incorrectly addressed letters is more than 3 .

1. Find the size of this test.
2. Find the probability of a Type II error in the case where \(p\) is in fact 0.10 Table 1 below gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]

   \(p\)	0.075	0.100	0.125	0.150	0.175	0.200	0.225
   Power	0.35	\(s\)	0.75	0.87	0.94	0.97	0.99

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
3. Write down the value of \(s\). A visiting consultant uses an alternative system to test the same hypotheses. A sample of 15 letters is taken. If these are all correctly addressed then \(\mathrm { H } _ { 0 }\) is accepted. If 2 or more are found to have been incorrectly addressed then \(\mathrm { H } _ { 0 }\) is rejected. If only one is found to be incorrectly addressed then a further random sample of 15 is taken and \(\mathrm { H } _ { 0 }\) is rejected if 2 or more are found to have been incorrectly addressed in this second sample, otherwise \(\mathrm { H } _ { 0 }\) is accepted.
4. Find the size of the test used by the consultant. \section*{Question 4 continues on page 8} For your convenience Table 1 is repeated here \begin{table}[h]

   \(p\)	0.075	0.100	0.125	0.150	0.175	0.200	0.225
   Power	0.35	\(s\)	0.75	0.87	0.94	0.97	0.99

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table} Figure 1 shows the graph of the power function of the test used by the consultant. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8dfc721d-4782-4482-9976-11189370f3b7-07_1712_1673_660_130} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure}
5. On Figure 1 draw the graph of the power function of the manager's test.
   (2)
6. State, giving your reasons, which test you would recommend.
   (2)

8 The random variable \(R\) has the distribution \(\mathrm { B } ( 10 , p )\). The null hypothesis \(\mathrm { H } _ { 0 } : p = 0.7\) is to be tested against the alternative hypothesis \(\mathrm { H } _ { 1 } : p < 0.7\), at a significance level of \(5 \%\).

1. Find the critical region for the test and the probability of making a Type I error.
2. Given that \(p = 0.4\), find the probability that the test results in a Type II error.
3. Given that \(p\) is equally likely to take the values 0.4 and 0.7 , find the probability that the test results in a Type II error.