2.05b Hypothesis test for binomial proportion

4. A proportion \(p\) of letters sent by a company are incorrectly addressed and if \(p\) is thought to be greater than 0.05 then action is taken.
Using \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\), a manager from the company takes a random sample of 40 letters and rejects \(\mathrm { H } _ { 0 }\) if the number of incorrectly addressed letters is more than 3 .

1. Find the size of this test.
2. Find the probability of a Type II error in the case where \(p\) is in fact 0.10 Table 1 below gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]

   \(p\)	0.075	0.100	0.125	0.150	0.175	0.200	0.225
   Power	0.35	\(s\)	0.75	0.87	0.94	0.97	0.99

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
3. Write down the value of \(s\). A visiting consultant uses an alternative system to test the same hypotheses. A sample of 15 letters is taken. If these are all correctly addressed then \(\mathrm { H } _ { 0 }\) is accepted. If 2 or more are found to have been incorrectly addressed then \(\mathrm { H } _ { 0 }\) is rejected. If only one is found to be incorrectly addressed then a further random sample of 15 is taken and \(\mathrm { H } _ { 0 }\) is rejected if 2 or more are found to have been incorrectly addressed in this second sample, otherwise \(\mathrm { H } _ { 0 }\) is accepted.
4. Find the size of the test used by the consultant. \section*{Question 4 continues on page 8} For your convenience Table 1 is repeated here \begin{table}[h]

   \(p\)	0.075	0.100	0.125	0.150	0.175	0.200	0.225
   Power	0.35	\(s\)	0.75	0.87	0.94	0.97	0.99

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table} Figure 1 shows the graph of the power function of the test used by the consultant. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8dfc721d-4782-4482-9976-11189370f3b7-07_1712_1673_660_130} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure}
5. On Figure 1 draw the graph of the power function of the manager's test.
   (2)
6. State, giving your reasons, which test you would recommend.
   (2)

3. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2 . The firm believes that the appointment of a new salesman will increase the number of houses sold. The firm tests its belief by recording the number of houses sold, \(x\), in the week following the appointment. The firm sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 2\) and \(\mathrm { H } _ { 1 } : \lambda > 2\), where \(\lambda\) is the mean number of houses sold per week, and rejects the null hypothesis if \(x \geqslant 3\)

1. Find the size of the test.
2. Show that the power function for this test is $$1 - \frac { 1 } { 2 } e ^ { - \lambda } \left( 2 + 2 \lambda + \lambda ^ { 2 } \right)$$ The table below gives the values of the power function to 2 decimal places. \begin{table}[h]

   \(\lambda\)	2.5	3.0	3.5	4.0	5.0	7.0
   Power	0.46	\(r\)	0.68	\(s\)	0.88	0.97

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
3. Calculate the values of \(r\) and \(s\).
4. Draw a graph of the power function on the graph paper provided on page 6
5. Find the range of values of \(\lambda\) for which the power of this test is greater than 0.6 For your convenience Table 1 is repeated here.

   \(\lambda\)	2.5	3.0	3.5	4.0	5.0	7.0
   Power	0.46	\(r\)	0.68	\(s\)	0.88	0.97

   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Table 1} \includegraphics[alt={},max width=\textwidth]{4f096806-33da-453f-a4c1-12be20d1a96d-06_2125_1603_614_166}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-07_72_47_2615_1886}

5. Water is tested at various stages during a purification process by an environmental scientist. A certain organism occurs randomly in the water at a rate of \(\lambda\) every 10 ml . The scientist selects a random sample of 20 ml of water to check whether there is evidence that \(\lambda\) is greater than 1 . The criterion the scientist uses for rejecting the hypothesis that \(\lambda = 1\) is that there are 4 or more organisms in the sample of 20 ml .

1. Find the size of the test.
2. When \(\lambda = 2.5\) find P (Type II error). A statistician suggests using an alternative test. The statistician's test involves taking a random sample of 10 ml and rejecting the hypothesis that \(\lambda = 1\) if 2 or more organisms are present but accepting the hypothesis if no organisms are in the sample. If only 1 organism is found then a second random sample of 10 ml is taken and the hypothesis is rejected if 2 or more organisms are present, otherwise the hypothesis is accepted.
3. Show that the power of the statistician's test is given by $$1 - \mathrm { e } ^ { - \lambda } - \lambda ( 1 + \lambda ) \mathrm { e } ^ { - 2 \lambda }$$ Table 1 below gives some values, to 2 decimal places, of the power function of the statistician's test. \begin{table}[h] \end{table} Table 1 Figure 1 shows a graph of the power function for the scientist's test.
   (e) On the same axes draw the graph of the power function for the statistician's test. Given that it takes 20 minutes to collect and test a 20 ml sample and 15 minutes to collect and test a 10 ml sample
   (f) show that the expected time of the statistician's test is slower than the scientist's test for \(\lambda \mathrm { e } ^ { - \lambda } > \frac { 1 } { 3 }\) (g) By considering the times when \(\lambda = 1\) and \(\lambda = 2\) together with the power curves in part (e) suggest, giving a reason, which test you would use.
   (2) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{399f7507-4878-45ad-b77e-02ebd807ed75-10_1185_1157_1452_392} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{399f7507-4878-45ad-b77e-02ebd807ed75-11_81_47_2622_1886}

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.

Information was collected about accidents on the Seapron bypass. It was found that the number of accidents per month could be modelled by a Poisson distribution with mean 2.5 Following some work on the bypass, the numbers of accidents during a series of 3-month periods were recorded. The data were used to test whether or not there was a change in the mean number of accidents per month.

1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, find the critical region for this test. You should state the probability in each tail.
2. State P(Type I error) using this test. Data from the series of 3-month periods are recorded for 2 years.
3. Find the probability that at least 2 of these 3-month periods give a significant result. Given that the number of accidents per month on the bypass, after the work is completed, is actually 2.1 per month,
4. find P (Type II error) for the test in part (a)

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).

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.

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}

Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is \(\frac { 1 } { 3 }\) Naasir and his friends play the game 15 times.

1. Find the probability that Naasir wins
   1. exactly 2 games,
   2. more than 5 games. Naasir claims he has a method to help him win more than \(\frac { 1 } { 3 }\) of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.
2. Stating your hypotheses clearly, test Naasir's claim at the \(5 \%\) level of significance.

5.

1. 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.
2. Write down the hypotheses that should be used to test the manager's suspicion.
3. 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
4. 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.
5. 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.
6. 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)}

A company sells seeds and claims that \(55 \%\) of its pea seeds germinate.

1. Write down a reason why the company should not justify their claim by testing all the pea seeds they produce. A random selection of the pea seeds is planted in 10 trays with 24 seeds in each tray.
2. Assuming that the company's claim is correct, calculate the probability that in at least half of the trays 15 or more of the seeds germinate.
3. Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution. A random sample of 240 pea seeds was planted and 150 of these seeds germinated.
4. Assuming that the company's claim is correct, use a normal approximation to find the probability that at least 150 pea seeds germinate.
5. Using your answer to part (d), comment on whether or not the proportion of the company's pea seeds that germinate is different from the company's claim of \(55 \%\)

7 Mohammed is conducting a medical trial to study the effect of two drugs, \(A\) and \(B\), on the amount of time it takes to recover from a particular illness. Drug \(A\) is used by one group of 60 patients and drug \(B\) is used by a second group of 60 patients. The results are summarised in the table:

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]

Jeevan thinks that a six-sided die is biased in favour of six. In order to test this, Jeevan throws the die 10 times. If the die shows a six on at least 4 throws out of 10, she will conclude that she is correct.

1. State appropriate null and alternative hypotheses. [1]
2. Calculate the probability of a Type I error. [3]
3. Explain what is meant by a Type II error in this situation. [1]
4. If the die is actually biased so that the probability of throwing a six is \(\frac{1}{3}\), calculate the probability of a Type II error. [3]

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]