Calculate probability of Type I error

7 In the past, the mean time for Jenny's morning run was 28.2 minutes. She does some extra training and she wishes to test whether her mean time has been reduced. After the training Jenny takes a random sample of 40 morning runs. She decides that if the sample mean run time is less than 27 minutes she will conclude that the training has been effective. You may assume that, after the training, Jenny's run time has a standard deviation of 4.0 minutes.

1. State suitable null and alternative hypotheses for Jenny's test.
2. Find the probability that Jenny will make a Type I error.
3. Jenny found that the sample mean run time was 27.2 minutes. Explain briefly whether it is possible for her to make a Type I error or a Type II error or both.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The manufacturer of a certain type of biscuit claims that \(10 \%\) of packets include a free offer printed on the packet. Jyothi suspects that the true proportion is less than \(10 \%\). He plans to test the claim by looking at 40 randomly selected packets and, if the number which include the offer is less than 2 , he will reject the manufacturer's claim.

1. State suitable hypotheses for the test.
2. Find the probability of a Type I error.
   On another occasion Jyothi looks at 80 randomly selected packets and finds that exactly 6 include the free offer.
3. Calculate an approximate \(90 \%\) confidence interval for the proportion of packets that include the offer.
4. Use your confidence interval to comment on the manufacturer's claim. \(6 X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

3 In the past, Arvinder has found that the mean time for his journey to work is 35.2 minutes. He tries a different route to work, hoping that this will reduce his journey time. Arvinder decides to take a random sample of 25 journeys using the new route. If the sample mean is less than 34.7 minutes he will conclude that the new route is quicker. Assume that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.

1. Find the probability that a Type I error occurs.
2. Arvinder finds that the sample mean is 34.5 minutes. Explain briefly why it is impossible for him to make a Type II error.

1. A machine fills cartons with juice.

The amount of juice in a carton is normally distributed with mean \(\mu \mathrm { ml }\) and standard deviation 8 ml . A manager wants to test whether or not the amount of juice in the cartons, \(X \mathrm { ml }\), is less than 330 ml . The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice \(\bar { x } \mathrm { ml }\).

1. Using a \(5 \%\) level of significance, find the critical region of \(\bar { X }\) for this test. State your hypotheses clearly. The Director is concerned about the machine filling the cartons with more than 330 ml of juice as well as less than 330 ml of juice. The Director takes a sample of 55 cartons, records the mean amount of juice \(\bar { y } \mathrm { ml }\) and uses a test with a critical region of $$\{ \bar { Y } < 328 \} \cup \{ \bar { Y } > 332 \}$$
2. Find P (Type I error) for the Director's test. When \(\mu = 325 \mathrm { ml }\)
3. find P (Type II error) for the test in part (a)

A train company claims that the probability \(p\) of one of its trains arriving late is 10\%. A regular traveller sets up the hypothesis \(H_0: p = 0.1\) and decides that the probability is greater than 10\% and decides to test this by randomly selecting 12 trains and recording the number \(X\) of trains that were late. The traveller sets up the hypotheses \(H_0: p = 0.1\) and \(H_1: p > 0.1\) and decides to reject \(H_0\) if \(x \ge 2\).

1. Find the size of the test. [1]
2. Show that the power function of the test is $$1 - (1 - p)^{10}(1 + 10p + 55p^2).$$ [4]
3. Calculate the power of the test when
   1. \(p = 0.2\),
   2. \(p = 0.6\). [3]
4. Comment on your results from part (c). [1]

Define

1. a Type I error, [1]
2. the size of a test. [1]

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 recorded. The probability of Jane stating the correct symbol is denoted by \(p\). To test the hypothesis H₀: \(p = 0.25\) against H₁: \(p > 0.25\), a critical region of \(X > 6\) is used.

1. [(c)] Find the size of this test. [3]
2. Show that the power function of this test is \(8p^7 - 7p^8\). [3]

Given that \(p = 0.3\), calculate

1. [(e)] the power of this test, [1]
2. the probability of a Type II error. [2]
3. Suggest two ways in which you might reduce the probability of a Type II error. [2]

(Total 12 marks)