2.05c Significance levels: one-tail and two-tail

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]

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]

A train company claims that the probability \(p\) of one of its trains arriving late is 10\%. A regular traveller on the company's trains believes 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 accepts the null hypothesis if \(x \leq 2\).

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]

Pierre is a chef. He claims that 90% of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5% significance level. [7]

Martin grows cucumbers from seed. In the past, he has found that 70% of all seeds successfully germinate and grow into cucumber plants. He decides to try out a new brand of seed. The producer of this brand claims that these seeds are more likely to successfully germinate than other brands of seeds. Martin sows 20 of this new brand of seed and 18 successfully germinate. Carry out a hypothesis test at the 5% level of significance to investigate the producer's claim. [7 marks]

It is known from historical data that 15% of the residents of a town buy the local weekly newspaper, 'Local News'. A new free weekly paper is introduced into the town. The owners of 'Local News' are interested to know whether the introduction of the free newspaper has changed the proportion of residents who buy their paper. In a random sample of 50 residents of the town taken after the free newspaper was introduced, it was found that 3 of them purchased 'Local News' regularly. Investigate, at the 5% significance level, whether this sample provides evidence that the proportion of local residents who buy 'Local News' has changed. [6 marks]

The proportion of vegans in a city is thought to be 8% The owner of an organic food café in this city believes that the proportion of their customers who are vegan is greater than 8% To test this belief, a random sample of 50 customers at the café were interviewed and it was found that 7 of them were vegan. Investigate, at the 5% level, whether this sample supports the owner's belief. [5 marks]

Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day. Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size \(n\), can be modelled by the binomial distribution B(\(n\), 0.20).

1. There are 10 students in Ellie's statistics class. Using the distributional model suggested by Ellie, find the probability that, of the students in her class:
   1. two or fewer eat at least five portions of fruit and vegetables every day; [1 mark]
   2. at least one but fewer than four eat at least five portions of fruit and vegetables every day; [2 marks]
2. Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day.
   1. Name the sampling method used by Declan. [1 mark]
   2. Describe one weakness of this sampling method. [1 mark]
   3. Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the 5\% significance level to investigate whether Declan's belief is supported by this evidence. [6 marks]

Suzanne is a member of a sports club. For each sport she competes in, she wins half of the matches.

1. After buying a new tennis racket Suzanne plays 10 matches and wins 7 of them. Investigate, at the 10% level of significance, whether Suzanne's new racket has made a difference to the probability of her winning a match. [7 marks]
2. After buying a new squash racket, Suzanne plays 20 matches. Find the minimum number of matches she must win for her to conclude, at the 10% level of significance, that the new racket has improved her performance. [5 marks]

James is playing a mathematical game on his computer. The probability that he wins is 0.6 As part of an online tournament, James plays the game 10 times. Let \(Y\) be the number of games that James wins.

1. State two assumptions, in context, for \(Y\) to be modelled as \(B(10, 0.6)\) [2 marks]
2. Find \(P(Y = 4)\) [1 mark]
3. Find \(P(Y \geq 4)\) [2 marks]
4. After practising the game, James claims that he has increased his probability of winning the game. In a random sample of 15 subsequent games, he wins 12 of them. Test at a 5% significance level whether James's claim is correct. [6 marks]

A council found that 70% of its new local businesses made a profit in their first year. The council introduced an incentive scheme for its residents to encourage the use of new local businesses. At the end of the scheme, a random sample of 25 new local businesses was selected and it was found that 21 of them had made a profit in their first year. Using a binomial distribution, investigate, at the 2.5% level of significance, whether there is evidence of an increase in the proportion of new local businesses making a profit in their first year. [6 marks]

It is known that 80% of all diesel cars registered in 2017 had carbon monoxide (CO) emissions less than 0.3 g/km. Talat decides to investigate whether the proportion of diesel cars registered in 2022 with CO emissions less than 0.3 g/km has **changed**. Talat will carry out a hypothesis test at the 10% significance level on a random sample of 25 diesel cars registered in 2022.

1. 1. State suitable null and alternative hypotheses for Talat's test. [1 mark]
   2. Using a 10% level of significance, find the critical region for Talat's test. [5 marks]
   3. In his random sample, Talat finds 18 cars with CO emissions less than 0.3 g/km. State Talat's conclusion in context. [1 mark]
2. Talat now wants to use his random sample of 25 diesel cars, registered in 2022, to investigate whether the proportion of diesel cars in England with CO emissions more than 0.5 g/km has changed from the proportion given by the Large Data Set. Using your knowledge of the Large Data Set, give **two** reasons why it is not possible for Talat to do this. [2 marks]

During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching. In January 2007, he increased his weekly exercise to try to improve his health. For the next 7 years, he only fell ill during 2 Christmas holidays.

1. Using a binomial distribution, investigate, at the 5% level of significance, whether there is evidence that John's rate of illness during the Christmas holidays had decreased since increasing his weekly exercise. [6 marks]
2. State two assumptions, regarding illness during the Christmas holidays, that are necessary for the distribution you have used in part (a) to be valid. For each assumption, comment, in context, on whether it is likely to be correct. [4 marks]

It is known that 20% of plants of a certain type suffer from a fungal disease, when grown under normal conditions. Some plants of this type are grown using a new method. A random sample of 250 of these plants is chosen, and it is found that 36 suffer from the disease. Test, at the 2% significance level, whether there is evidence that the new method reduces the proportion of plants which suffer from the disease. [7]

Casey and Riley attend a large school. They are discussing the music preferences of the students at their school. Casey believes that the favourite band of 75% of the students is Blue Rocking. Riley believes that the true figure is greater than 75%. They plan to carry out a hypothesis test at the 5% significance level, using the hypotheses \(H_0: p = 0.75\) and \(H_1: p > 0.75\). They choose a random sample of 60 students from the school, and note the number, \(X\), who say that their favourite band is Blue Rocking. They find that \(X = 50\).

1. Assuming the null hypothesis to be true, Riley correctly calculates that \(P(X = 50) = 0.0407\), correct to 3 significant figures. Riley says that, because this value is less than 0.05, the null hypothesis should be rejected. Explain why this statement is incorrect. [1]
2. Carry out the test. [5]
3. 1. State which mathematical model is used in the calculation in part (b), including the value(s) of any parameter(s). [1]
   2. The random sample was chosen without replacement. Explain whether this invalidates the model used in part (b). [1]

In this question you must show detailed reasoning. Research showed that in May 2017 62% of adults over 65 years of age in the UK used a certain online social media platform. Later in 2017 it was believed that this proportion had increased. In December 2017 a random sample of 59 adults over 65 years of age in the UK was collected. It was found that 46 of the 59 adults used this online social media platform. Use a suitable hypothesis test to determine whether there is evidence at the 1% level to suggest that the proportion of adults over 65 in the UK who used this online social media platform had increased from May 2017 to December 2017. [7]

Records from the 1950s showed that 35\% of human babies were born without wisdom teeth. It is believed that as part of the evolutionary process more babies are now born without wisdom teeth. In a random sample of 140 babies, collected in 2020, a researcher found that 61 were born without wisdom teeth. The researcher made the following statement. ``This shows that the percentage of babies born without wisdom teeth has increased from 35\%.''

1. Explain whether this statement can be fully justified. [1]
2. Conduct a hypothesis test at the 5\% level to determine whether there is any evidence to suggest that more than 35\% of babies are now born without wisdom teeth. [7]

Edward can correctly identify 20% of types of wild flower. He studies some books to see if he can improve how often he can correctly identify types of wild flower. He collects a random sample of 10 types of wild flower in order to test whether or not he has improved.

1. 1. Write suitable hypotheses for this test.
   2. State a suitable test statistic that he could use. [2]
2. Using a 5% level of significance, find the critical region for this test. [3]
3. State the probability of a Type I error for this test and explain what it means in this context. [2]
4. Edward correctly identifies 4 of the 10 types of wild flower he collected. What conclusion should Edward reach? [2]