Hypothesis test of binomial distributions

3. A coin is tossed 20 times, giving 16 heads.

1. Test at the \(1 \%\) significance level whether the coin is fair, stating your hypotheses clearly.
2. Find the critical region for the same test at the \(0.1 \%\) significance level.

7 Items from a production line are examined for any defects. The probability that any item will be found to be defective is 0.15 , independently of all other items.

1. A batch of 16 items is inspected. Using tables of cumulative binomial probabilities, or otherwise, find the probability that
   (a) at least 4 items in the batch are defective,
   (b) exactly 4 items in the batch are defective.
2. Five batches, each containing 16 items, are taken.
   (a) Find the probability that at most 2 of these 5 batches contain at least 4 defective items.
   (b) Find the expected number of batches that contain at least 4 defective items.

2. A test statistic has a distribution \(\mathrm { B } ( 25 , p )\). Given that $$\mathrm { H } _ { 0 } : p = 0.5 \quad \mathrm { H } _ { 1 } : p \neq 0.5$$

1. find the critical region for the test statistic such that the probability in each tail is as close as possible to \(2.5 \%\).
2. State the probability of incorrectly rejecting \(\mathrm { H } _ { 0 }\) using this critical region.

5 To test whether a coin is biased or not, it is tossed 10 times. The coin will be considered biased if there are 9 or 10 heads, or 9 or 10 tails.

1. Show that the probability of making a Type I error in this test is approximately 0.0215 .
2. Find the probability of making a Type II error in this test when the probability of a head is actually 0.7.

9 Briony suspects that a particular 6-sided dice is biased in favour of 2. She plans to throw the dice 35 times and note the number of times that it shows a 2 . She will then carry out a test at the \(4 \%\) significance level. Find the rejection region for the test.

3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.

1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
3. Determine the expected number of answers that Sarah will get right.
4. Find the probability that Sarah gets more than 25 correct answers out of 30.

1. Briefly explain what is meant by
   1. a statistical model,
      (2 marks)
   2. a sampling frame,
   3. a sampling unit.
   4. (a) Explain what is meant by the critical region of a statistical test.
   5. Under a hypothesis \(\mathrm { H } _ { 0 }\), an event \(A\) can happen with probability \(4 \cdot 2 \%\). The event \(A\) does then happen. State, with justification, whether \(\mathrm { H } _ { 0 }\) should be accepted or rejected at the \(5 \%\) significance level.

5

1. Deng wishes to test whether a certain coin is biased so that it is more likely to show Heads than Tails. He throws it 12 times. If it shows Heads more than 9 times, he will conclude that the coin is biased. Calculate the significance level of the test.
2. Deng throws another coin 100 times in order to test, at the \(5 \%\) significance level, whether it is biased towards Heads. Find the rejection region for this test.

2 A die has six faces numbered \(1,2,3,4,5,6\). Manjit suspects that the die is biased so that it shows a six on fewer throws than it would if it were fair. In order to test her suspicion, she throws the die a certain number of times and counts the number of sixes.

1. State suitable null and alternative hypotheses for Manjit's test.
2. There are no sixes in the first 15 throws. Show that this result is not significant at the \(5 \%\) level.
3. Find the smallest value of \(n\) such that, if there are no sixes in the first \(n\) throws, this result is significant at the 5\% level.

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 .

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.

18

1. State the Null and Alternative hypotheses for this test. 18
2. State, in context, the conclusion to this test. 18 It is believed that 25\% of the customers at a bakery buy a loaf of bread. is believed that \(25 \%\) of the customers at a bakery buy a loaf of bread.
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2 A shop obtains apples from a certain farm. It has been found that 5\% of apples from this farm are Grade A. Following a change in growing conditions at the farm, the shop management plan to carry out a hypothesis test to find out whether the proportion of Grade A apples has increased. They select 25 apples at random. If the number of Grade A apples is more than 3 they will conclude that the proportion has increased.

1. State suitable null and alternative hypotheses for the test.
2. Find the probability of a Type I error.
   In fact 2 of the 25 apples were Grade A .
3. Which of the errors, Type I or Type II, is possible? Justify your answer.

7 In a research laboratory where plants are studied, the probability of a certain type of plant surviving was 0.35 . The laboratory manager changed the growing conditions and wished to test whether the probability of a plant surviving had increased.

1. The plants were grown in rows, and when the manager requested a random sample of 8 plants to be taken, the technician took all 8 plants from the front row. Explain what was wrong with the technician’s sample.
2. A suitable sample of 8 plants was taken and 4 of these 8 plants survived. State whether the manager's test is one-tailed or two-tailed and also state the null and alternative hypotheses. Using a \(5 \%\) significance level, find the critical region and carry out the test.
3. State the meaning of a Type II error in the context of the test in part (ii).
4. Find the probability of a Type II error for the test in part (ii) if the probability of a plant surviving is now 0.4.

6 Stephan is an athlete who competes in the high jump. In the past, Stephan has succeeded in \(90 \%\) of jumps at a certain height. He suspects that his standard has recently fallen and he decides to carry out a hypothesis test to find out whether he is right. If he succeeds in fewer than 17 of his next 20 jumps at this height, he will conclude that his standard has fallen.

1. Find the probability of a Type I error.
2. In fact Stephan succeeds in 18 of his next 20 jumps. Which of the errors, Type I or Type II, is possible? Explain your answer.

19 Martin grows cucumbers from seed.

3 The manufacturer of electronic components uses the following process to test the proportion of defective items produced. A random sample of 20 is taken from a large batch of components.

• If no defective item is found, the batch is accepted.
• If two or more defective items are found, the batch is rejected.
• If one defective item is found, a second random sample of 20 is taken. If two or more defective items are found in this second sample, the batch is rejected, otherwise the batch is accepted.

The proportion of defective items in the batch is denoted by \(p\), and \(q = 1 - p\).

1. Show that the probability that a batch is accepted is \(q ^ { 20 } + 20 p q ^ { 38 } ( q + 20 p )\). For a particular component, \(p = 0.01\).
2. Given that a batch is accepted, find the probability that it is accepted as a result of the first sample.

4. A sweet shop produces different coloured sweets and sells them in bags. The proportion of green sweets produced is \(p\) Each bag is filled with a random sample of \(n\) sweets. The mean number of green sweets in a bag is 4.2 and the variance is 3.57

1. Find the value of \(n\) and the value of \(p\) The proportion of red sweets produced by the shop is 0.35
2. Find the probability that, in a random sample of 25 sweets, the number of red sweets exceeds the expected number of red sweets. The shop claims that \(10 \%\) of its customers buy more than two bags of sweets. A random sample of 40 customers is taken and 1 customer buys more than two bags of sweets.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy more than two bags of sweets is less than the shop's claim. State your hypotheses clearly.
   

3. A single observation \(x\) is to be taken from a Binomial distribution \(\mathrm { B } ( 20 , p )\). This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)

1. Using a \(5 \%\) level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to \(2.5 \%\).
2. State the actual significance level of this test. The actual value of \(x\) obtained is 3 .
3. State a conclusion that can be drawn based on this value giving a reason for your answer.

3 The proportion of adults in a large village who support a proposal to build a bypass is denoted by \(p\). A random sample of size 20 is selected from the adults in the village, and the members of the sample are asked whether or not they support the proposal.

1. Name the probability distribution that would be used in a hypothesis test for the value of \(p\).
2. State the properties of a random sample that explain why the distribution in part (i) is likely to be a good model.
   \(4 X\) is a continuous random variable.

1 A random variable has the distribution \(\mathrm { B } ( n , p )\). It is required to test \(\mathrm { H } _ { 0 } : p = \frac { 2 } { 3 }\) against \(\mathrm { H } _ { 1 } : p < \frac { 2 } { 3 }\) at a significance level as close to \(1 \%\) as possible, using a sample of size \(n = 8,9\) or 10 . Use tables to find which value of \(n\) gives such a test, stating the critical region for the test and the corresponding significance level.
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