Calculate Type I error probability

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.

6 When a child completes an online exercise called a Mathlit, they might be awarded a medal. The publishers claim that the probability that a randomly chosen child who completes a Mathlit will be awarded a medal is \(\frac { 1 } { 3 }\). Asha wishes to test this claim. She decides that if she is awarded no medals while completing 10 Mathlits, she will conclude that the true probability is less than \(\frac { 1 } { 3 }\).

1. Use a binomial distribution to find the probability of a Type I error.
   The true probability of being awarded a medal is denoted by \(p\).
2. Given that the probability of a Type II error is 0.8926 , find the value of \(p\).

7 Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number \(X\) of days on which she sees it. She models the distribution of \(X\) by \(\mathrm { B } ( 31 , p )\), where \(p\) is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that \(p = 0.3\), but in 2022 Rita suspected that the value of \(p\) had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.

1. Use the binomial distribution to test at the \(5 \%\) significance level whether Rita's suspicion is justified.
   In July 2023, she noted the value of \(X\) and carried out another test at the \(5 \%\) significance level using the same hypotheses.
2. Calculate the probability of a Type I error.
   Rita models the number of sightings, \(Y\), per year of a different, very rare, kind of bird by the distribution \(B ( 365,0.01 )\).
3. 1. Use a suitable approximating distribution to find \(\mathrm { P } ( Y = 4 )\).
   2. Justify your approximating distribution in this context.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 Marie claims that she can predict the winning horse at the local races. There are 8 horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next 10 races and, if she is correct in 3 or more races, Nadine will accept Marie's claim.

1. State suitable null and alternative hypotheses.
2. Calculate the probability of a Type I error.
3. State the significance level of the test.

4 It is claimed that 1 in every 4 packets of certain biscuits contains a free gift. Marisa and André both suspect that the true proportion is less than 1 in 4.

1. Marisa chooses 20 packets at random. She decides that if fewer than 3 contain free gifts, she will conclude that the claim is not justified. Use a binomial distribution to find the probability of a Type I error.
2. André chooses 25 packets at random. He decides to carry out a significance test at the \(1 \%\) level, using a binomial distribution. Given that only 1 of the 25 packets contains a free gift, carry out the test.

6 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked. A random sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the \(5 \%\) significance level.

1. Explain why the test is one-tailed and state suitable null and alternative hypotheses.
2. Use a binomial distribution to find the critical region, and find the probability of a Type I error.
3. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.

4 In a certain city it is necessary to pass a driving test in order to be allowed to drive a car. The probability of passing the driving test at the first attempt is 0.36 on average. A particular driving instructor claims that the probability of his pupils passing at the first attempt is higher than 0.36 . A random sample of 8 of his pupils showed that 7 passed at the first attempt.

1. Carry out an appropriate hypothesis test to test the driving instructor's claim, using a significance level of \(5 \%\).
2. In fact, most of this random sample happened to be careful and sensible drivers. State which type of error in the hypothesis test (Type I or Type II) could have been made in these circumstances and find the probability of this type of error when a sample of size 8 is used for the test.

7 Leila suspects that a particular six-sided die is biased so that the probability, \(p\), that it will show a six is greater than \(\frac { 1 } { 6 }\). She tests the die by throwing it 5 times. If it shows a six on 3 or more throws she will conclude that it is biased.

1. State what is meant by a Type I error in this situation and calculate the probability of a Type I error.
2. Assuming that the value of \(p\) is actually \(\frac { 2 } { 3 }\), calculate the probability of a Type II error. Leila now throws the die 80 times and it shows a six on 50 throws.
3. Calculate an approximate \(96 \%\) confidence interval for \(p\).

3 Sumitra has a six-sided die. She suspects that it is biased so that it shows a six less often than it would if it were fair. She decides to test the die by throwing it 30 times and noting the number of throws on which it shows a six.

1. It shows a six on exactly 2 throws. Use a binomial distribution to carry out the test at the \(5 \%\) significance level.
2. Later, Sumitra repeats the test at the \(5 \%\) significance level by throwing the die 30 times again. Find the probability of a Type I error in this second test.

5 Every month Susan enters a particular lottery. The lottery company states that the probability, \(p\), of winning a prize is 0.0017 each month. Susan thinks that the probability of winning is higher than this, and carries out a test based on her 12 lottery results in a one-year period. She accepts the null hypothesis \(p = 0.0017\) if she has no wins in the year and accepts the alternative hypothesis \(p > 0.0017\) if she wins a prize in at least one of the 12 months.

1. Find the probability of the test resulting in a Type I error.
2. If in fact the probability of winning a prize each month is 0.0024 , find the probability of the test resulting in a Type II error.
3. Use a suitable approximation, with \(p = 0.0024\), to find the probability that in a period of 10 years Susan wins a prize exactly twice.

3 Joshi suspects that a certain die is biased so that the probability of showing a six is less than \(\frac { 1 } { 6 }\). He plans to throw the die 25 times and if it shows a six on fewer than 2 throws, he will conclude that the die is biased in this way.

1. Find the probability of a Type I error and state the significance level of the test. Joshi now decides to throw the die 100 times. It shows a six on 9 of these throws.
2. Calculate an approximate \(95 \%\) confidence interval for the probability of showing a six on one throw of this die.

6 At the last election, 70\% of people in Apoli supported the president. Luigi believes that the same proportion support the president now. Maria believes that the proportion who support the president now is \(35 \%\). In order to test who is right, they agree on a hypothesis test, taking Luigi's belief as the null hypothesis. They will ask 6 people from Apoli, chosen at random, and if more than 3 support the president they will accept Luigi's belief.

1. Calculate the probability of a Type I error.
2. If Maria's belief is true, calculate the probability of a Type II error.
3. In fact 2 of the 6 people say that they support the president. State which error, Type I or Type II, might be made. Explain your answer.

5 It is known that when seeds of a certain type are planted, on average \(10 \%\) of the resulting plants reach a height of 1 metre. A gardener wishes to investigate whether a new fertiliser will increase this proportion. He plants a random sample of 18 seeds of this type, using the fertiliser, and notes how many of the resulting plants reach a height of 1 metre.

1. In fact 4 of the 18 plants reach a height of 1 metre. Carry out a hypothesis test at the \(8 \%\) significance level.
2. Explain which of the errors, Type I or Type II, might have been made in part (i). Later, the gardener plants another random sample of 18 seeds of this type, using the fertiliser, and again carries out a hypothesis test at the \(8 \%\) significance level.
3. Find the probability of a Type I error.

5 It is claimed that \(30 \%\) of packets of Froogum contain a free gift. Andre thinks that the actual proportion is less than \(30 \%\) and he decides to carry out a hypothesis test at the \(5 \%\) significance level. He buys 20 packets of Froogum and notes the number of free gifts he obtains.

1. State null and alternative hypotheses for the test.
2. Use a binomial distribution to find the probability of a Type I error. Andre finds that 3 of the 20 packets contain free gifts.
3. Carry out the test.

9 The random variable \(A\) has the distribution \(\mathrm { B } ( 30 , p )\). A test is carried out of the hypotheses \(\mathrm { H } _ { 0 } : p = 0.6\) against \(\mathrm { H } _ { 1 } : p < 0.6\). The critical region is \(A \leqslant 13\).

1. State the probability that \(\mathrm { H } _ { 0 }\) is rejected when \(p = 0.6\).
2. Find the probability that a Type II error occurs when \(p = 0.5\).
3. It is known that on average \(p = 0.5\) on one day in five, and on other days the value of \(p\) is 0.6 . On each day two tests are carried out. If the result of the first test is that \(\mathrm { H } _ { 0 }\) is rejected, the value of \(p\) is adjusted if necessary, to ensure that \(p = 0.6\) for the rest of the day. Otherwise the value of \(p\) remains the same as for the first test. Calculate the probability that the result of the second test is to reject \(\mathrm { H } _ { 0 }\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

5. A statistician believes a coin is biased and the probability, \(p\), of getting a head when the coin is tossed is less than 0.5 The statistician decides to test this by tossing the coin 10 times and recording the number, \(X\), of heads. He sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) and rejects the null hypothesis if \(x < 3\)

1. Find the size of the test.
2. Show that the power function of this test is $$( 1 - p ) ^ { 8 } \left( 36 p ^ { 2 } + 8 p + 1 \right)$$ Table 1 gives values, to 2 decimal places, of the power function for the statistician's test. \begin{table}[h] \end{table} Table 1
   (d) On the axes below draw the graph of the power function for the statistician's test.
   (e) Find the range of values of \(p\) for which the probability of accepting the coin as unbiased, when in fact it is biased, is less than or equal to 0.4 \includegraphics[max width=\textwidth, alt={}, center]{1d84c9fc-be67-45be-b439-3111c48ff1cb-09_1143_1209_945_402}

3. A certain vaccine is known to be only \(70 \%\) effective against a particular virus; thus \(30 \%\) of those vaccinated will actually catch the virus. In order to test whether or not a new and more expensive vaccine provides better protection against the same virus, a random sample of 30 people were chosen and given the new vaccine. If fewer than 6 people contracted the virus the new vaccine would be considered more effective than the current one.

1. Write down suitable hypotheses for this test.
2. Find the probability of making a Type I error.
3. Find the power of this test if the new vaccine is
   1. \(80 \%\) effective,
   2. \(90 \%\) effective. An independent research organisation decided to test the new vaccine on a random sample of 50 people to see if it could be considered more than \(70 \%\) effective. They required the probability of a Type I error to be as close as possible to 0.05 .
4. Find the critical region for this test.
5. State the size of this critical region.
6. Find the power of this test if the new vaccine is
   1. \(80 \%\) effective,
   2. \(90 \%\) effective.
7. Give one advantage and one disadvantage of the second test.

12 A firm claims that no more than \(2 \%\) of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than \(2 \%\). In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the \(5 \%\) significance level, using the null hypothesis \(p = 0.02\).

1. Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
2. The researcher finds that 18 out of the 600 packets are underweight. A colleague says
   " 18 out of 600 is \(3 \%\), so there is evidence that the actual proportion of underweight bags is greater than \(2 \%\)." Criticise this statement.

2 A binomial hypothesis test was carried out at the \(5 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.6 \\ & \mathrm { H } _ { 1 } : p > 0.6 \end{aligned}$$ A sample of size 30 was used to carry out the test.
Find the probability that a Type I error was made.
Circle your answer.
[0pt] [1 mark] \(4.4 \%\) 4.8\% 5.0\% 9.4\%

At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.1. The hospital's business model assumed that this probability will be reduced. They wish to test whether this probability is now less than 0.1. A random sample of 50 appointments is selected and the number of patients that did not arrive is noted. This figure is used as a test statistic at the 5\% significance level.

1. Explain why this test is a one-tailed test and state suitable null and alternative hypotheses. [2]
2. Use a binomial distribution to find the critical region and find the probability of a Type I error. [5]
3. In fact 3 patients out of the 50 did not arrive. State the conclusion of the test, explaining your answer. [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]

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 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]