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

A single observation x is to be taken from a Poisson distribution with parameter \(\lambda\). This observation is to be used to test H₀: \(\lambda\) = 7 against H₁: \(\lambda\) ≠ 7.

1. Using a 5\% significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to 2.5\%. [5]
2. Write down the significance level of this test. [1]

The actual value of x obtained was 5.

1. State a conclusion that can be drawn based on this value. [2]

A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that 40\% of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

1. Define the population associated with the magazine. [1]
2. Suggest a suitable sampling frame for the survey. [1]
3. Identify the sampling units. [1]
4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. [2]

As a pilot study the editor took a random sample of 25 subscribers.

1. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. [3]

In fact only 6 subscribers agreed to the name being changed.

1. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the percentage agreeing to the change is less that the editor believes. [5]

The full survey is to be carried out using 200 randomly chosen subscribers.

1. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. [7]

A doctor expects to see, on average, 1 patient per week with a particular disease.

1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer. [3]
2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. [4]

The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.

1. Test, at the 5\% level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. [6]

Medical research into the nature of the disease discovers that it can be passed from one patient to another.

1. Explain whether or not this research supports your choice of model. Give a reason for your answer. [2]

Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.

1. Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. [2]

Find the probability that in any randomly selected 10 minute interval

1. exactly 6 cars pass this point, [3]
2. at least 9 cars pass this point. [2]

After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.

1. Test, at the 5\% level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly. [6]

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]

A single observation \(x\) is to be taken from a Binomial distribution B(20, \(p\)). This observation is used to test \(H_0 : p = 0.3\) against \(H_1 : p \neq 0.3\)

1. Using a 5\% level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to 2.5\%. [3]
2. State the actual significance level of this test. [2]

The actual value of \(x\) obtained is 3.

1. State a conclusion that can be drawn based on this value giving a reason for your answer. [2]

A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.

1. 1. Test, at the 10\% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
   2. State the minimum number of visits required to obtain a significant result.
   [7]
2. State an assumption that has been made about the visits to the server. [1]

In a random two minute period on a Saturday the web server is visited 20 times.

1. Using a suitable approximation, test at the 10\% level of significance, whether or not the rate of visits is greater on a Saturday. [6]

A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the 5\% level of significance, test whether or not there is evidence to reject the teacher's claim. State your hypotheses clearly. [6]

Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the 5\% level of significance. State your hypotheses clearly. [6]

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 was 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]

Brad planted 25 seeds in his greenhouse. He has read in a gardening book that the probability of one of these seeds germinating is 0.25. Ten of Brad's seeds germinated. He claimed that the gardening book had underestimated this probability. Test, at the 5% level of significance, Brad's claim. State your hypotheses clearly. [7]

1. Explain what you understand by a critical region of a test statistic. [2]

The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac{1}{7}\).

1. Find the probability that on a particular day there are fewer than 2 breakdowns. [3]
2. Find the probability that during a 14-day period there are at most 4 breakdowns. [3]

The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.

1. Test, at the 5% level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly. [7]

It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.

1. Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to 2.5\% as possible. [6]
2. State the actual significance level of the above test. [1]

At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.

1. Test, at the 10\% level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly. [7]

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]
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]
3. Find the actual significance level of this test. [2]

In the sample of 50 the actual number of faulty bolts was 8.

1. Comment on the company's claim in the light of this value. Justify your answer. [2]

The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.

1. Test at the 1\% level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly. [6]

A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:

1. Show that the mean number of errors per page in this sample of pages is 2. [2]
2. Find the variance of the number of errors per page in this sample. [2]
3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution. [1]

Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2.

1. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a 5\% level of significance. [6]

In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.

1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]

Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.

1. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. State the significance level of this test giving your answer to 2 significant figures. [1]

An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.

1. Three days are chosen at random. Find the probability that the pollution level is
   1. low on all 3 days, [2]
   2. low on at least one day, [2]
   3. low on one day, medium on another day, and high on the other day. [3]
2. Ten days are chosen at random. Find the probability that
   1. there are no days when the pollution level is high, [2]
   2. there is exactly one day when the pollution level is high. [3]

The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.

1. Carry out a test at the 5% level to determine if there is evidence to suggest that she is correct. Use hypotheses \(H_0: p = 0.5\), \(H_1: p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(H_1\) has this form. [5]

Mark is playing solitaire on his computer. The probability that he wins a game is 0.2, independently of all other games that he plays.

1. Find the expected number of wins in 12 games. [2]
2. Find the probability that
   1. he wins exactly 2 out of the next 12 games that he plays, [3]
   2. he wins at least 2 out of the next 12 games that he plays. [3]
3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the 5\% level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test. [9]

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]

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]

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]