One-tailed test critical region

7 A national survey shows that \(95 \%\) of year 12 students use social media. Arvin suspects that the percentage of year 12 students at his college who use social media is less than the national percentage. He chooses a random sample of 20 students at his college and notes the number who use social media. He then carries out a test at the \(2 \%\) significance level.

1. Find the rejection region for the test.
2. Find the probability of a Type I error.
3. Jimmy believes that the true percentage at Arvin's college is \(70 \%\). Assuming that Jimmy is correct, find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A machine is supposed to produce random digits. Bob thinks that the machine is not fair and that the probability of it producing the digit 0 is less than \(\frac { 1 } { 10 }\). In order to test his suspicion he notes the number of times the digit 0 occurs in 30 digits produced by the machine. He carries out a test at the \(10 \%\) significance level.

1. State suitable null and alternative hypotheses.
2. Find the rejection region for the test.
3. State the probability of a Type I error.
   It is now given that the machine actually produces a 0 once in every 40 digits, on average.
4. Find the probability of a Type II error.
5. Explain the meaning of a Type II error in this context.

4 A cereal manufacturer claims that \(25 \%\) of cereal packets contain a free gift. Lola suspects that the true proportion is less than \(25 \%\). In order to test the manufacturer's claim at the \(5 \%\) significance level, she checks a random sample of 20 packets.

1. Find the critical region for the test.
2. Hence find the probability of a Type I error. Lola finds that 2 packets in her sample contain a free gift.
3. State, with a reason, the conclusion she should draw.

3 It is known that \(25 \%\) of students in a particular city are smokers. A random sample of 20 of the students is selected.

1. (A) Find the probability that there are exactly 4 smokers in the sample.
   (B) Find the probability that there are at least 3 but no more than 6 smokers in the sample
   (C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Explain why the alternative hypothesis has the form that it does
3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
4. In fact there are 3 smokers in the sample. Complete the test, stating your conclusion clearly.

3 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.

1. (A) Find the probability that there are exactly 2 faulty tiles in the sample.
   (B) Find the probability that there are more than 2 faulty tiles in the sample.
   (C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Explain why the alternative hypothesis has the form that it does.
3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
4. In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.

3 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, \(20 \%\) of pupils eat at least five a day.

1. 15 children are selected at random.
   (A) Find the probability that exactly 3 of them eat at least five a day.
   (B) Find the probability that at least 3 of them eat at least five a day.
   (C) Find the expected number who eat at least five a day. A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Give a reason for your choice of the alternative hypothesis.
3. Find the critical region for the test at the \(10 \%\) significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.

9 A pharmaceutical company is developing a new drug to treat a certain disease. The company will continue to develop the drug if the proportion \(p\) of those who have the disease and show a substantial improvement after treatment is greater than 0.7 . The company carries out a test, at the \(5 \%\) significance level, on a random sample of 14 patients who suffer from the disease.

1. Find the critical region for the test.
2. Given that 12 of the 14 patients in the sample show a substantial improvement, carry out the test.
3. Find the probability that the test results in a Type II error if in fact \(p = 0.8\). RECOGNISING ACHIEVEMENT

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.

10 A company operates trains. The company claims that \(92 \%\) of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.

1. Assuming that \(92 \%\) of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
   1. exactly 28 trains arrive on time,
   2. more than 27 trains arrive on time. A journalist believes that the percentage of trains operated by this company which arrive on time is lower than \(92 \%\).
2. To investigate the journalist's belief a hypothesis test will be carried out at the \(1 \%\) significance level. A random sample of 18 trains is selected. For this hypothesis test,

1. Sam and Tessa are testing a spinner to see if the probability, p , of it landing on red is less than \(\frac { 1 } { 5 }\). They both use a \(10 \%\) significance level.

Sam decides to spin the spinner 20 times and record the number of times it lands on red.

1. Find the critical region for Sam's test.
2. Write down the size of Sam's test. Tessa decides to spin the spinner until it lands on red and she records the number of spins.
3. Find the critical region for Tessa's test.
4. Find the size of Tessa's test.
5. 1. Show that the power function for Sam's test is given by $$( 1 - p ) ^ { 19 } ( 1 + 19 p )$$
   2. Find the power function for Tessa's test.
6. With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when \(\mathrm { p } = 0.15\)

4 The proportion, \(p\), of people in a particular town who use the local supermarket is unknown. A random sample of 30 people in the town is taken and each person is asked if they use the local supermarket. The manager of the supermarket claims that 35\% of the people in the town use the local supermarket. The random sample is used to conduct a hypothesis test at the \(5 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.35 \\ & \mathrm { H } _ { 1 } : p \neq 0.35 \end{aligned}$$ Show that the probability that a Type I error is made is 0.0356 , correct to four decimal places.

A student has an ordinary six-sided dice. The student suspects that it is biased against six, so that when it is thrown, it is less likely to show a six than if it were fair. In order to test this suspicion, the student plans to carry out a hypothesis test at the 5% significance level. The student throws the dice 100 times and notes the number of times, \(X\), that it shows a six.

1. Determine the largest value of \(X\) that would provide evidence at the 5% significance level that the dice is biased against six. [3]

Later another student carries out a similar test, at the 5% significance level. This student also throws the dice 100 times.

1. It is given that the dice is fair. Find the probability that the conclusion of the test is that there is significant evidence that the dice is biased against six. [1]

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below.

Type of defect	Colour	Fabric	Sewing	Sizing
Probability	0.25	0.30	0.40	0.05

Shirts with defects are packed in boxes of 30 at random.

1. Find the probability that:
   1. a box contains exactly 5 shirts with a colour defect [2 marks]
   2. a box contains fewer than 15 shirts with a sewing defect [2 marks]
   3. a box contains at least 20 shirts which do not have a fabric defect. [3 marks]
2. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses H\(_0\): \(p = 0.3\) H\(_1\): \(p < 0.3\) She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.
   1. Using a 5\% level of significance, find the critical region for \(x\). [5 marks]
   2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]

A company produces sweets of varying colours. The company claims that the proportion of blue sweets is 13·6%. A consumer believes that the true proportion is less than this. In order to test this belief, the consumer collects a random sample of 80 sweets.

1. State suitable hypotheses for the test. [1]
2. 1. Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, 5%.
   2. Given that there are 6 blue sweets in the sample of 80, complete the significance test. [5]
3. Suppose the proportion of blue sweets claimed by the company is correct. The consumer conducts the sampling and testing process on a further 20 occasions, using the sample size of 80 each time. What is the expected number of these occasions on which the consumer would reach the incorrect conclusion? [2]
4. Now suppose that the proportion of blue sweets is 7%. Find the probability of a Type II error. Interpret your answer in context. [3]

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses $$H_0 : p = 0.3$$ $$H_1 : p < 0.3$$ She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.

1. Using a 5% level of significance, find the critical region for \(x\). [5 marks]
2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]