Two-tailed test critical region

6 A factory makes chocolates of different types. The proportion of milk chocolates made on any day is denoted by \(p\). It is desired to test the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.8\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p < 0.8\). The test consists of choosing a random sample of 25 chocolates. \(\mathrm { H } _ { 0 }\) is rejected if the number of milk chocolates is \(k\) or fewer. The test is carried out at a significance level as close to \(5 \%\) as possible.

1. Use tables to find the value of \(k\), giving the values of any relevant probabilities.
2. The test is carried out 20 times, and each time the value of \(p\) is 0.8 . Each of the tests is independent of all the others. State the expected number of times that the test will result in rejection of the null hypothesis.
3. The test is carried out once. If in fact the value of \(p\) is 0.6 , find the probability of rejecting \(\mathrm { H } _ { 0 }\).
4. The test is carried out twice. Each time the value of \(p\) is equally likely to be 0.8 or 0.6 . Find the probability that exactly one of the two tests results in rejection of the null hypothesis.

1. Past information shows that \(25 \%\) of adults in a large population have a particular allergy.

Rylan believes that the proportion that has the allergy differs from 25\%
He takes a random sample of 50 adults from the population.
Rylan carries out a test of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\) using a \(5 \%\) level of significance.

1. Write down the alternative hypothesis for Rylan's test.
2. Find the critical region for this test. You should state the probability associated with each tail, which should be as close to \(2.5 \%\) as possible.
3. State the actual probability of incorrectly rejecting \(\mathrm { H } _ { 0 }\) for this test. Rylan finds that 10 of the adults in his sample have the allergy.
4. State the conclusion of Rylan's hypothesis test.

1. The proportion of left-handed adults in a country is \(10 \%\)

Freya believes that the proportion of left-handed adults under the age of 25 in this country is different from 10\% She takes a random sample of 40 adults under the age of 25 from this country to investigate her belief.

1. Find the critical region for a suitable test to assess Freya's belief. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   • state the probability of rejection in each tail
   • Write down the actual significance level of your test in part (a)
   In Freya's sample 7 adults were left-handed.
2. With reference to your answer in part (a) comment on Freya's belief.

12 A survey conducted in 2021 showed that 10\% of British adults were vegetarians. A dietitian believes that the proportion of British adults who are vegetarians may have changed, so decides to conduct a hypothesis test at the \(5 \%\) level of significance. In a random sample of 112 adults, the dietitian finds that there are 19 vegetarians. Carry out the hypothesis test to determine whether there is any evidence to support the dietitian's belief.

5. A company claims that \(35 \%\) of its peas germinate. In order to test this claim Ann decides to plant 15 of these peas and record the number which germinate.

1. 1. State suitable hypotheses for a two-tailed test of this claim.
   2. Using a \(5 \%\) level of significance, find an appropriate critical region for this test. The probability in each of the tails should be as close to \(2.5 \%\) as possible.
2. Ann found that 8 of the 15 peas germinated. State whether or not the company's claim is supported. Give a reason for your answer.
3. State the actual significance level of this test.

7. A manufacturer produces packets of sweets. Each packet contains 25 sweets. The manufacturer claims that, on average, 40\% of the sweets in each packet are red. A packet is selected at random.

1. Using a \(1 \%\) level of significance, find the critical region for a two-tailed test that the proportion of red sweets is 0.40 You should state the probability in each tail, which should be as close as possible to 0.005
2. Find the actual significance level of this test. The manufacturer changes the production process to try to reduce the number of red sweets. She chooses 2 packets at random and finds that 8 of the sweets are red.
3. Test, at the \(1 \%\) level of significance, whether or not there is evidence that the manufacturer's changes to the production process have been successful. State your hypotheses clearly.

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   Q7

1. A company produces packets of sunflower seeds. Each packet contains 40 seeds. The company claims that, on average, only 35\% of its sunflower seeds do not germinate.

A packet is selected at random.

1. Using a \(5 \%\) level of significance, find an appropriate critical region for a two-tailed test that the proportion of sunflower seeds that do not germinate is 0.35 You should state your hypotheses clearly and state the probability, which should be as close as possible to \(2.5 \%\), for each tail of your critical region.
2. Write down the actual significance level of this test. Past records suggest that \(2.8 \%\) of the company's sunflower seeds grow to a height of more than 3 metres.
   A random sample of 250 of the company's sunflower seeds is taken and 11 of them grow to a height of more than 3 metres.
3. Using a suitable approximation test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than \(2.8 \%\) State your hypotheses clearly.

6. A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.

1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.
2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac { 1 } { 4 }\). The probability of rejection in either tail should be as close as possible to 0.025
3. Find the actual significance level of this test. In the sample of 50 the actual number of faulty bolts was 8 .
4. Comment on the company's claim in the light of this value. Justify your answer. The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
5. Test at the \(1 \%\) level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.

7. A teacher thinks that \(20 \%\) of the pupils in a school read the Deano comic regularly. He chooses 20 pupils at random and finds 9 of them read the Deano.

1. 1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20\%. State your hypotheses clearly.
   2. State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the \(5 \%\) level.
      (9) The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
2. By combining all 5 samples and using a suitable approximation test, at the \(5 \%\) level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20\%.
3. Comment on your results for the tests in part (a) and part (b).

3. In a sack containing a large number of beads \(\frac { 1 } { 4 }\) are coloured gold and the remainder are of different colours. A group of children use some of the beads in a craft lesson and do not replace them. Afterwards the teacher wishes to know whether or not the proportion of gold beads left in the sack has changed. He selects a random sample of 20 beads and finds that 2 of them are coloured gold. Stating your hypotheses clearly test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of gold beads has changed.

4. Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager questions whether or not there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.

1. Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's question. You should state the probability of rejection in each tail which should be less than 0.05 .
2. Write down the actual significance level of a test based on your critical region from part (a). The manager found that 11 customers from the sample of 20 had bought baked beans in single tins.
3. Comment on this finding in the light of your critical region found in part (a).

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.

6. In a manufacturing process \(25 \%\) of articles are thought to be defective. Articles are produced in batches of 20

1. A batch is selected at random. Using a \(5 \%\) significance level, find the critical region for a two tailed test that the probability of an article chosen at random being defective is 0.25
   You should state the probability in each tail which should be as close as possible to 0.025 The manufacturer changes the production process to try to reduce the number of defective articles. She then chooses a batch at random and discovers there are 3 defective articles.
2. Test at the \(5 \%\) level of significance whether or not there is evidence that the changes to the process have reduced the percentage of defective articles. State your hypotheses clearly.

4. From past records a manufacturer of glass vases knows that \(15 \%\) of the production have slight defects. To monitor the production, a random sample of 20 vases is checked each day and the number of vases with slight defects is recorded.

1. Using a 5\% significance level, find the critical regions for a two-tailed test of the hypothesis that the probability of a vase with slight defects is 0.15 . The probability of rejecting, in either tail, should be as close as possible to \(2.5 \%\).
2. State the actual significance level of the test described in part (a). A shop sells these vases at a rate of 2.5 per week. In the 4 weeks of December the shop sold 15 vases.
3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales per week had increased in December.
   (6 marks)

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.

3. The random variable \(X\) is modelled by a binomial distribution \(\mathrm { B } ( n , p )\), with \(n = 20\) and \(p\) unknown. It is suspected that \(p = 0 \cdot 4\).

1. Find the critical region for the test of \(\mathrm { H } _ { 0 } : p = 0.4\) against \(\mathrm { H } _ { 1 } : p \neq 0.4\), at the \(5 \%\) significance level.
2. Find the critical region if, instead, the alternative hypothesis is \(\mathrm { H } _ { 1 } : p < 0.4\).

3. A primary school teacher finds that exactly half of his year 6 class have mobile phones. He decides to investigate whether the proportion of pupils with mobile phones is different from 0.5 in the year 5 class at his school. There are 25 pupils in the year 5 class.

1. State the hypotheses that he should use.
2. Find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
3. Determine the significance level of this test. He finds that eight of the year 5 pupils have mobile phones and concludes that there is not sufficient evidence of the proportion being different from 0.5
4. Stating the new hypotheses clearly, find if the number of year 5 pupils with mobile phones would have been significant if he had tested whether or not the proportion was less than 0.5 and used the largest critical region with a probability of less than \(5 \%\).
   (3 marks)

3. A jar contains a large number of sweets which have either soft centres or hard centres. The jar is thought to contain equal proportions of sweets with soft centres and sweets with hard centres. A random sample of 20 sweets is taken from the jar and the number of sweets with hard centres is recorded.

1. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that there are equal proportions of sweets with soft centres and sweets with hard centres in the jar.
2. Calculate the probability of a Type I error for this test. Given that there are 3 times as many sweets with soft centres as there are sweets with hard centres,
3. calculate the probability of a Type II error for this test.

3 The manufacturers of a brand of chocolates claim that, on average, \(30 \%\) of their chocolates have hard centres. In a random sample of 8 chocolates from this manufacturer, 5 had hard centres. Test, at the \(5 \%\) significance level, whether there is evidence that the population proportion of chocolates with hard centres is not \(30 \%\), stating your hypotheses clearly. Show the values of any relevant probabilities.

In a sack containing a large number of beads \(\frac{1}{4}\) are coloured gold and the remainder are of different colours. A group of children use some of the beads in a craft lesson and do not replace them. Afterwards the teacher wishes to know whether or not the proportion of gold beads left in the sack has changed. She selects a random sample of 20 beads and finds that 2 of them are coloured gold. Stating your hypotheses clearly test, at the 10\% level of significance, whether or not there is evidence that the proportion of gold beads has changed. [7]

Past records show that 20\% of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps were taken and 2 of them had bought them in single packets.

1. Use these data to test, at the 5\% level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly. [6]

At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03. To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.

1. Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. Write down the significance level of this test. [1]

From past records a manufacturer of ceramic plant pots knows that 20\% of them will have defects. To monitor the production process, a random sample of 25 pots is checked each day and the number of pots with defects is recorded.

1. Find the critical regions for a two-tailed test of the hypothesis that the probability that a plant pot has defects is 0.20. The probability of rejection in either tail should be as close as possible to 2.5\%. [5]
2. Write down the significance level of the above test. [1]

A garden centre sells these plant pots at a rate of 10 per week. In an attempt to increase sales, the price was reduced over a six-week period. During this period a total of 74 pots was sold.

1. Using a 5\% level of significance, test whether or not there is evidence that the rate of sales per week has increased during this six-week period. [7]