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

12 In the past, the time spent by customers in a certain shop had mean 10.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 12.0 minutes.

1. Assuming that the standard deviation is unchanged, test at the \(1 \%\) significance level whether the mean time spent by customers in the shop has changed.
2. Another random sample of 50 customers is chosen and a similar test at the \(1 \%\) significance level is carried out. Given that the population mean time has not changed, state the probability that the conclusion of the test will be that the population mean time has changed.

12 It is known that under the standard treatment for a certain disease, \(9.7 \%\) of patients with the disease experience side effects within one year. In a trial of a new treatment, 450 patients with this disease were selected and the number, \(X\), that experienced side effects within one year was noted. It was found that 51 of the 450 patients experienced side effects within one year.

1. Test, at the \(10 \%\) significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.
2. It was later discovered that all 450 patients selected for the trial were treated in the same hospital. Comment on the validity of the model used in part (a).

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.

16

State, in context, two assumptions that are necessary for the distribution that you have used in part (a) to be valid.

18 It is known from previous data that 14\% of the visitors to a particular cookery website are under 30 years of age. To encourage more visitors under 30 years of age a large advertising campaign took place to target this age group. To test whether the campaign was effective, a sample of 60 visitors to the website was selected. It was found that 15 of the visitors were under 30 years of age. 18

1. Explain why a one-tailed hypothesis test should be used to decide whether the sample provides evidence that the campaign was effective. 18
2. Carry out the hypothesis test at the \(5 \%\) level of significance to investigate whether the sample provides evidence that the proportion of visitors under 30 years of age has increased.
   18
3. State one necessary assumption about the sample for the distribution used in part (b) to be valid.
   [0pt] [1 mark]

16 It is believed that a coin is biased so that the probability of obtaining a head when the coin is tossed is 0.7 16

1. Assume that the probability of obtaining a head when the coin is tossed is indeed 0.7
   16
   1. (i) Find the probability of obtaining exactly 6 heads from 7 tosses of the coin.
      16
   2. (ii) Find the mean number of heads obtained from 7 tosses of the coin.
      16
   3. Harry believes that the probability of obtaining a head for this coin is actually greater than 0.7 To test this belief he tosses the coin 35 times and obtains 28 heads. Carry out a hypothesis test at the \(10 \%\) significance level to investigate Harry's belief. \includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-24_2492_1721_217_150}
      \includegraphics[max width=\textwidth, alt={}]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-28_2498_1722_213_147}

5. (a) The discrete random variable \(X \sim \mathrm {~B} ( 40,0.27 )\) $$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$ Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that 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.
(b) Write down the hypotheses that should be used to test the manager's suspicion.
(c) Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05
(d) Find the actual significance level of a test based on your critical region from part (c). One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.
(e) Comment on the manager's suspicion in the light of this observation. Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.
(f) Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.

2. The discrete random variable \(X \sim \mathrm {~B} ( 30,0.28 )\)

1. Find \(\mathrm { P } ( 5 \leq X < 12 )\). Past records from a large supermarket show that \(25 \%\) of people who buy eggs, buy organic eggs. On one particular day a random sample of 40 people is taken from those that had bought eggs and 16 people are found to have bought organic eggs.
2. Test, at the \(1 \%\) significance level, whether or not the proportion \(p\) of people who bought organic eggs that day had increased. State your hypotheses clearly.
3. State the conclusion you would have reached if a \(5 \%\) significance level had been used for this test. \section*{(Total for Question 2 is 8 marks)}

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.

4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects \(2 p\) coins that are less than 6.12 grams or greater than 8.12 grams.

1. The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Weight of UK two pence coins} \includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
   \end{figure} Explain why the weight of 2 p coins can be modelled using a normal distribution.
2. Assume the distribution of the weight of \(2 p\) coins is normally distributed. Calculate the proportion of \(2 p\) coins that are rejected by this brand of coin counting machine.
3. A manager suspects that a large batch of \(2 p\) coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.

   Summary statistics
   Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
   6.89	0.296	6.45	6.63	6.88	7.08	7.48

   i) What assumption must be made about the weights of coins in this batch in order to conduct a test of significance on the sample mean? State, with a reason, whether you think this assumption is reasonable.
   ii) Assuming the population standard deviation is 0.357 grams, test at the \(1 \%\) significance level whether the mean weight of the \(2 p\) coins in this batch is less than 7.12 grams.

At a certain large school it was found that the proportion of students not wearing correct uniform was 0.15. The school sent a letter to parents asking them to ensure that their children wear the correct uniform. The school now wishes to test whether the proportion not wearing correct uniform has been reduced.

1. It is suggested that a random sample of the students in Grade 12 should be used for the test. Give a reason why this would not be an appropriate sample. [1]
2. State suitable null and alternative hypotheses. [1]
3. Use a binomial distribution to find the probability of a Type I error. You must justify your answer fully. [5]
4. In fact 4 students out of the 50 are not wearing correct uniform. State the conclusion of the test, explaining your answer. [2]
5. State, with a reason, which of the errors, Type I or Type II, may have been made. [2]

A suitable sample of 50 students is selected and the number not wearing correct uniform is noted. This figure is used to carry out a test at the 5% significance level.

The number of faults in cloth made on a certain machine has a Poisson distribution with mean 2.4 per 10 m\(^2\). An adjustment is made to the machine. It is required to test at the 5% significance level whether the mean number of faults has decreased. A randomly selected 30 m\(^2\) of cloth is checked and the number of faults is found.

1. State suitable null and alternative hypotheses for the test. [1]
2. Find the probability of a Type I error. [3]

Exactly 3 faults are found in the randomly selected 30 m\(^2\) of cloth.

1. Carry out the test at the 5% significance level. [2]

Later a similar test was carried out at the 5% significance level, using another randomly selected 30 m\(^2\) of cloth.

1. Given that the number of faults actually has a Poisson distribution with mean 0.5 per 10 m\(^2\), find the probability of a Type II error. [2]

In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution Po(0.31). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5, carry out the test at the 2.5% significance level. [5]

The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.

1. A hypothesis test at the 10% level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test. [5]
2. Find the probability of a Type I error. [2]

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]

The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1

1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods. [2]
2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. [2]

The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.

1. Use the tables to find the value of \(w\) [3]

A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.

1. State the null hypothesis for this test. [1]
2. Determine the critical region for the test at the 5\% level of significance. [2]

A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution. The fisherman takes 5 fishing trips each lasting 1 hour.

1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. [6]

The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,

1. carry out the test at the 5\% level of significance. State your hypotheses clearly. [6]

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]

The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week. Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is

1. exactly 4, [2]
2. more than 5. [2]

Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.

1. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. [3]

The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.

1. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased. [5]

The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2.5. The firm appoints a new salesman and wants to find out whether or not house sales increase as a result. After the appointment of the salesman, the number of house sales in a randomly chosen 4-week period is 14. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the new salesman has increased house sales. [7]

The owner of a small restaurant decides to change the menu. A trade magazine claims that 40\% of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.

1. Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners X who choose organic foods. [2]
2. Find P(5 < X < 15). [4]
3. Find the mean and standard deviation of X. [3]

The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.

1. Test, at the 5\% level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly. [5]

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]