2.05b Hypothesis test for binomial proportion

Any patient who fails to turn up for an outpatient appointment at a hospital is described as a 'no-show'. At a particular hospital, on average 15% of patients are no-shows. A random sample of 20 patients who have outpatient appointments is selected.

1. Find the probability that
   1. there is exactly 1 no-show in the sample, [3]
   2. there are at least 2 no-shows in the sample. [2]

The hospital management introduces a policy of telephoning patients before appointments. It is hoped that this will reduce the proportion of no-shows. In order to check this, a random sample of \(n\) patients is selected. The number of no-shows in the sample is recorded and a hypothesis test is carried out at the 5% level.

1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
2. In the case that \(n = 20\) and the number of no-shows in the sample is 1, carry out the test. [4]
3. In another case, where \(n\) is large, the number of no-shows in the sample is 6 and the critical value for the test is 8. Complete the test. [3]
4. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the 5% level. [2]

It is known that on average 85% of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.

1. 1. Find the probability that exactly 12 germinate. [3]
   2. Find the probability that fewer than 12 germinate. [2]

The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting \(n\) of these seeds and sowing them. He then carries out a hypothesis test at the 1% significance level to investigate whether he is correct.

1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
2. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test. [4]
3. Suppose instead that Ramesh conducts the trial with \(n = 50\), and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35, complete the test. [3]
4. If \(n\) is small, there is no point in carrying out the test at the 1% significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of \(n\) for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer. [3]

An insurance company conducts its business by using a Call Centre. The average number of calls per minute is 3.5. In the first minute after a TV advertisement is shown, the number of calls received is 7.

1. Stating your hypotheses carefully, and working at the 5\% significance level, test whether the advertisement has had an effect. [5 marks]
2. Find the number of calls that would be required in the first minute for the null hypothesis to be rejected at the 0.1\% significance level. [3 marks]

A greengrocer sells apples from a barrel in his shop. He claims that no more than 5\% of the apples are of poor quality. When he takes 10 apples out for a customer, 2 of them are bad.

1. Stating your hypotheses clearly, test his claim at the 1\% significance level. [5 marks]
2. State an assumption that has been made about the selection of the apples. [1 mark]
3. When five other customers also buy 10 apples each, the numbers of bad apples they get are 1, 3, 1, 2 and 1 respectively. By combining all six customers' results, and using a suitable approximation, test at the 1\% significance level whether the combined results provide evidence that the proportion of bad apples in the barrel is greater than 5\%. [5 marks]
4. Comment briefly on your results in parts (a) and (c). [1 mark]

A certain type of lettuce seed has a 12\% failure rate for germination. In a new sample of 25 genetically modified seeds, only 1 did not germinate. Clearly stating your hypotheses, test, at the 1\% significance level, whether the GM seeds are better. [6 marks]

A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5.

1. Find P\((X = 0)\). [1 mark]
2. In 100 measurements, the value 0 occurs three times. Find the highest significance level at which you should reject the original hypothesis in favour of \(\lambda < 5\). [8 marks]

A small opinion poll shows that the Trendies have a \(10\%\) lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a \(10\%\) lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.

1. At the \(5\%\) significance level, test which is right, stating your null hypothesis carefully. [6 marks]
2. If it is indeed true that the Trendies are supported by \(55\%\) of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes. [8 marks]

The number of customers arriving at a store between 8.50 am and 9 am on Saturday mornings is a random variable which can be modelled by the distribution Po(11.0). Following a series of price cuts, on one particular Saturday morning 19 customers arrive between 8.50 am and 9 am. The store's management claims, first, that the mean number of customers has increased, and second, that this is due to the price cuts.

1. Test the first part of the claim, at the 5% significance level. [7]
2. Comment on the second part of the claim. [1]

The random variable \(R\) has the distribution B(10, \(p\)). The null hypothesis H\(_0\): \(p = 0.7\) is to be tested against the alternative hypothesis 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. [3]
2. Given that \(p = 0.4\), find the probability that the test results in a Type II error. [3]
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. [2]

The number of fruit pips in 1 cubic centimetre of raspberry jam has the distribution Po(\(\lambda\)). Under a traditional jam-making process it is known that \(\lambda = 6.3\). A new process is introduced and a random sample of 1 cubic centimetre of jam produced by the new process is found to contain 2 pips. Test, at the 5% significance level, whether this is evidence that under the new process the average number of pips has been reduced. [8]

55% of the pupils in a large school are girls. A member of the student council claims that the probability that a girl rather than a boy becomes Head Student is greater than 0.55. As evidence for his claim he says that 6 of the last 8 Head Students have been girls.

1. Use an exact binomial distribution to test the claim at the 10% significance level. [7]
2. A statistics teacher says that considering only the last 8 Head Students may not be satisfactory. Explain what needs to be assumed about the data for the test to be valid. [1]

The random variable \(R\) has the distribution Po\((\lambda)\). A significance test is carried out at the 1% level of the null hypothesis H\(_0\): \(\lambda = 11\) against H\(_1\): \(\lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]

A teacher wants to investigate the sports played by students at her school in their free time. She decides to ask a random sample of 120 pupils to complete a short questionnaire.

1. Give two reasons why the teacher might choose to use a sample survey rather than a census. [2 marks]
2. Suggest a suitable sampling frame that she could use. [1 mark]

The teacher believes that 1 in 20 of the students play tennis in their free time. She uses the data collected from her sample to test if the proportion is different from this.

1. Using a suitable approximation and stating the hypotheses that she should use, find the critical region for this test. The probability for each tail of the region should be as close as possible to 5\%. [6 marks]
2. State the significance level of this test. [1 mark]

A shoe shop sells on average 4 pairs of shoes per hour on a weekday morning.

1. Suggest a suitable distribution for modelling the number of sales made per hour on a weekday morning and state the value of any parameters needed. [1 mark]
2. Explain why this model might have to be modified for modelling the number of sales made per hour on a Saturday morning. [1 mark]
3. Find the probability that on a weekday morning the shop sells
   1. more than 4 pairs in a one-hour period,
   2. no pairs in a half-hour period,
   3. more than 4 pairs during each hour from 9 am until noon. [6 marks]

The area manager visits the shop on a weekday morning, the day after an advert appears in a local paper. In a one-hour period the shop sells 7 pairs of shoes, leading the manager to believe that the advert has increased the shop's sales.

1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence of an increase in sales following the appearance of the advert. [4 marks]

Six standard dice with faces numbered 1 to 6 are thrown together. Assuming that the dice are fair, find the probability that

1. none of the dice show a score of 6, [3 marks]
2. more than one of the dice shows a score of 6, [4 marks]
3. there are equal numbers of odd and even scores showing on the dice. [3 marks]

One of the dice is suspected of being biased such that it shows a score of 6 more often than the other numbers. This die is thrown eight times and gives a score of 6 three times.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not this die is biased towards scoring a 6. [7 marks]

A rugby player scores an average of 0.4 tries per match in which he plays.

1. Find the probability that he scores 2 or more tries in a match. [5 marks]

The team's coach moves the player to a different position in the team believing he will then score more frequently. In the next five matches he scores 6 tries.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of an increase in the mean number of tries the player scores per match as a result of playing in a different position. [5 marks]

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]

It is suggested that a Poisson distribution with parameter \(\lambda\) can model the number of currants in a currant bun. A random bun is selected in order to test the hypotheses H₀: \(\lambda = 8\) against H₁: \(\lambda \neq 8\), using a 10\% level of significance.

1. Find the critical region for this test, such that the probability in each tail is as close as possible to 5\%. [5]
2. Given that \(\lambda = 10\), find
   1. the probability of a type II error,
   2. the power of the test. [4]

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)

Rolls of cloth delivered to a factory contain defects at an average rate of 2 per metre. A quality assurance manager selects a random sample of 15 metres of cloth from each delivery to test whether or not there is evidence that \(\lambda > 0.3\). The criterion that the manager uses for rejecting the hypothesis that \(\lambda = 0.3\) is that there are 9 or more defects in the sample.

1. Find the size of the test. [2]

Table 1 gives some values, to 2 decimal places, of the power function of this test. \includegraphics{figure_5}

1. [(b)] Find the value of \(r\). [2]

The manager would like to design a test, of whether or not \(\lambda > 0.3\), that uses a smaller length of cloth. He chooses a length of 10 m and requires the probability of a type I error to be less than 10\%.

1. [(c)] Find the criterion to reject the hypothesis that \(\lambda = 0.3\) which makes the test as powerful as possible. [2]
2. Hence state the size of this second test. [1]

Table 2 gives some values, to 2 decimal places, of the power function for the test in part (c). \includegraphics{figure_5_table2}

1. [(e)] Find the value of \(s\). [2]
2. Using the same axes, on graph paper draw the graphs of the power functions of these two tests. [4]
3. [(g)] State the value of \(\lambda\) where the graphs cross.
   1. Explain the significance of \(\lambda\) where the graphs cross. [2]

There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm. Deliveries of cloth with \(\lambda = 0.3\) are unusable.

1. [(h)] Suggest, giving your reasons, which test manager should adopt. [2]

A drug is claimed to produce a cure to a certain disease in 35\% of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.

1. Write down suitable hypotheses to carry out this test. [2]
2. Find the probability of making a Type I error. [3] The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of \(p\) where \(p\) is the probability of the drug curing a person with the disease.

   P(cure)	0.2	0.3	0.4	0.5
   P(Type II error)	0.5880	\(r\)	0.8565	\(s\)

3. Calculate the value of \(r\) and the value of \(s\). [3]
4. Calculate the power of the test for \(p = 0.2\) and \(p = 0.4\) [2]
5. Comment, giving your reasons, on the suitability of this test procedure. [2]

Define, in terms of H\(_0\) and/or H\(_1\),

1. the size of a hypothesis test, [1]
2. the power of a hypothesis test. [1]

The probability of getting a head when a coin is tossed is denoted by \(p\). This coin is tossed 12 times in order to test the hypotheses H\(_0\): \(p = 0.5\) against H\(_1\): \(p \neq 0.5\), using a 5\% level of significance.

1. Find the largest critical region for this test, such that the probability in each tail is less than 2.5\%. [4]
2. Given that \(p = 0.4\)
   1. find the probability of a type II error when using this test,
   2. find the power of this test.
   [4]
3. Suggest two ways in which the power of the test can be increased. [2]

A manager in a sweet factory believes that the machines are working incorrectly and the proportion \(p\) of underweight bags of sweets is more than 5\%. He decides to test this by randomly selecting a sample of 5 bags and recording the number \(X\) that are underweight. The manager sets up the hypotheses H\(_0\): \(p = 0.05\) and H\(_1\): \(p > 0.05\) and rejects the null hypothesis if \(x > 1\).

1. Find the size of the test. [2]
2. Show that the power function of the test is $$1 - (1-p)^4(1+4p)$$ [3]

The manager goes on holiday and his deputy checks the production by randomly selecting a sample of 10 bags of sweets. He rejects the hypothesis that \(p = 0.05\) if more than 2 underweight bags are found in the sample.

1. Find the probability of a Type I error using the deputy's test. [2]

Question 3 continues on page 12 The table below gives some values, to 2 decimal places, of the power function for the deputy's test.

\(p\)	0.10	0.15	0.20	0.25
Power	0.07	\(s\)	0.32	0.47

1. Find the value of \(s\). [1]

The graph of the power function for the manager's test is shown in Figure 1. \includegraphics{figure_1}

1. On the same axes, draw the graph of the power function for the deputy's test. [1]
2. (i) State the value of \(p\) where these graphs intersect. (ii) Compare the effectiveness of the two tests if \(p\) is greater than this value. [2]

The deputy suggests that they should use his sampling method rather than the manager's.

1. Give a reason why the manager might not agree to this change. [1]

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 H\(_0\): \(p = 0.05\) and H\(_1\): \(p > 0.05\), a manager from the company takes a random sample of 40 letters and rejects H\(_0\) if the number of incorrectly addressed letters is more than 3.

1. Find the size of this test. [2]
2. Find the probability of a Type II error in the case where \(p\) is in fact 0.10 [2]

Table 1 below gives some values, to 2 decimal places, of the power function of this test.

\(p\)	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

Table 1

1. Write down the value of \(s\). [1]

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 H\(_0\) is accepted. If 2 or more are found to have been incorrectly addressed then H\(_0\) is rejected. If only one is found to be incorrectly addressed then a further random sample of 15 is taken and H\(_0\) is rejected if 2 or more are found to have been incorrectly addressed in this second sample, otherwise H\(_0\) is accepted.

1. Find the size of the test used by the consultant. [3]

Question 4 continues on page 8 \includegraphics{figure_1}

1. On Figure 1 draw the graph of the power function of the manager's test. [2]
2. State, giving your reasons, which test you would recommend. [2]

A proportion \(p\) of the items produced by a factory is defective. A quality assurance manager selects a random sample of 5 items from each batch produced to check whether or not there is evidence that \(p\) is greater than 0.10. The criterion that the manager uses for rejecting the hypothesis that \(p\) is 0.10 is that there are more than 2 defective items in the sample.

1. Find the size of the test. [2]

Table 1 gives some values, to 2 decimal places, of the power function of this test.

\(p\)	0.15	0.20	0.25	0.30	0.35	0.40
Power	0.03	\(r\)	0.10	0.16	0.24	0.32

1. Find the value of \(r\). [3]

One day the manager is away and an assistant checks the production by random sample of 10 items from each batch produced. The hypothesis that \(p = 0.10\) is rejected if more than 4 defectives are found in the sample.

1. Find P(Type I error) using the assistant's test. [2]

Table 2 gives some values, to 2 decimal places, of the power function for this test.

\(p\)	0.15	0.20	0.25	0.30	0.35	0.40
Power	0.01	0.03	0.08	0.15	0.25	\(s\)

1. Find the value of \(s\). [1]
2. Using the same axes, draw the graphs of the power functions of these two tests. [4]
3. 1. State the value of \(p\) where these graphs cross.
   2. Explain the significance if \(p\) is greater than this value.
   [2]

The manager studies the graphs in part \((e)\) but decides to carry on using the test based on a sample of size 5.

1. Suggest 2 reasons why the manager might have made this decision. [2]