Explain Type I or II error

2 Anton believes that \(10 \%\) of students at his college are left-handed. Aliya believes that this is an underestimate. She plans to carry out a hypothesis test of the null hypothesis \(p = 0.1\) against the alternative hypothesis \(p > 0.1\), where \(p\) is the actual proportion of students at the college that are left-handed. She chooses a random sample of 20 students from the college. She will reject the null hypothesis if at least 5 of these students are left-handed.

1. Explain what is meant by a Type I error in this context.
2. Find the probability of a Type I error in the test.
3. Given that the true value of \(p\) is 0.3 , find the probability of a Type II error in the test.

6

1. Explain what is meant by
   (a) a Type I error,
   (b) a Type II error.
2. Roger thinks that a box contains 6 screws and 94 nails. Felix thinks that the box contains 30 screws and 70 nails. In order to test these assumptions they decide to take 5 items at random from the box and inspect them, replacing each item after it has been inspected, and accept Roger's hypothesis (the null hypothesis) if all 5 items are nails.
   (a) Calculate the probability of a Type I error.
   (b) If Felix's hypothesis (the alternative hypothesis) is true, calculate the probability of a Type II error.

2 A manufacturer claims that \(20 \%\) of sugar-coated chocolate beans are red. George suspects that this percentage is actually less than \(20 \%\) and so he takes a random sample of 15 chocolate beans and performs a hypothesis test with the null hypothesis \(p = 0.2\) against the alternative hypothesis \(p < 0.2\). He decides to reject the null hypothesis in favour of the alternative hypothesis if there are 0 or 1 red beans in the sample.

1. With reference to this situation, explain what is meant by a Type I error.
2. Find the probability of a Type I error in George's test.

2. (a) Define

1. a Type I error,
2. a Type II error. A manufacturer sells socks in boxes of 50 .
   The mean number of faulty socks per box is 7.5 . In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.
   (b) (i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a \(5 \%\) significance level.
3. Stating your hypotheses clearly, carry out the test in part (i).
   (c) Find the probability of the Type I error for this test.
   (d) Given that the true mean number of faulty socks per box on the new machine is 5 , calculate the probability of a Type II error for this test.
   (e) Explain what would have been the effect of changing the significance level for the test in part (b) to \(2 \frac { 1 } { 2 } \%\).

5. Define

1. a Type I error,
2. the size of a test. Jane claims that she can read Alan's mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan's mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times \(X\) that Jane is correct is recorded. The probability of Jane stating the correct symbol is denoted by \(p\).
   To test the hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\) against \(\mathrm { H } _ { 1 } : p > 0.25\), a critical region of \(X > 6\) is used.
3. Find the size of this test.
4. Show that the power function of this test is \(8 p ^ { 7 } - 7 p ^ { 8 }\). Given that \(p = 0.3\), calculate
5. the power of this test,
6. the probability of a Type II error.
7. Suggest two ways in which you might reduce the probability of a Type II error.