7. A software company develops computer programs.
Its Lovelace program is found to glitch 7 times on average when it is run.

The company makes some alterations to improve the program.
It then tests to see if it has been successful in reducing the average number of glitches per run.
The glitches can be assumed to occur independently.

The altered program is actually found to glitch 5 times on average when run.

Due to the high value of the probability of a Type II error in part (b), the company makes an entirely new version of the program.
The new version of the program is designed to have an average of fewer than \(k\) glitches per run.

The random variable \(T\) represents the number of runs of the program until a run with no glitches occurs.

If the new version has an average of \(\lambda\) glitches per run, this new version of the program will be tested using the hypotheses

The company will reject \(\mathrm{H}_0\) if the first run without any glitches of the new version occurs within \(n\) runs.

The company requires \(\mathrm{P}(\text{Type II error}) \lt 0.2\)
Assuming that the new version of the program has an average of 2.75 glitches per run,

5. Some of the components produced by a factory are defective. The management requires that no more than 3% of the components produced are defective.
Niluki monitors the production process and takes a random sample of \(n\) components.

Niluki defines the random variable \(D_n\) to represent the number of defective components in a sample of size \(n\). She considers two tests A and B

In test A, Niluki uses \(n = 100\) and if \(D_{100} \geqslant 5\) she rejects \(\mathrm{H}_0\)

In test B, Niluki uses \(n = 80\) and

• if \(D_{80} \geqslant 5\) she rejects \(\mathrm{H}_0\)
• if \(D_{80} \leqslant 3\) she does not reject \(\mathrm{H}_0\)
• if \(D_{80} = 4\) she takes a second random sample of size 80 and if \(D_{80} \geqslant 1\) in this second sample then she rejects \(\mathrm{H}_0\) otherwise she does not reject \(\mathrm{H}_0\)

Given that the actual proportion of defective components is 0.06

Given also that, when the actual proportion of defective components is 0.06, the power of test B is 0.713

2. The number of errors made by a secretary is modelled by a Poisson distribution with a mean of 2.4 per 100 words.

A 100-word piece of work completed by the secretary is selected at random.

After a long holiday, a randomly selected piece of work containing 250 words completed by the secretary is examined to see if the rate of errors has changed.

5. A machine fills cartons with juice.

The amount of juice in a carton is normally distributed with mean \(\mu\) ml and standard deviation 8 ml.

A manager wants to test whether or not the amount of juice in the cartons, \(X\) ml, is less than 330 ml. The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice \(\bar{x}\) ml.

The Director is concerned about the machine filling the cartons with more than 330 ml of juice as well as less than 330 ml of juice. The Director takes a sample of 55 cartons, records the mean amount of juice \(\bar{y}\) ml and uses a test with a critical region of

When \(\mu = 325\) ml

7. A machine fills bags with flour. The weight of flour delivered by the machine into a bag, \(X\) grams, is normally distributed with mean \(\mu\) grams and standard deviation 30 grams.
To check if there is any change to the mean weight of flour delivered by the machine into each bag, Olaf takes a random sample of 10 bags. The weight of flour, \(x\) grams, in each bag is recorded and \(\bar{x} = 1020\)

Olaf decides to alter the test so that the hypotheses are \(\mathrm{H}_0: \mu = 1000\) and \(\mathrm{H}_1: \mu \gt 1000\) but keeps the level of significance at 5%

He takes a second sample of size \(n\) and finds the critical region, \(\overline{X} \gt c\)

When the true value of \(\mu\) is 1020 grams, the probability of making a Type II error is 0.0050, to 2 significant figures.

7. A manufacturer has a machine that produces lollipop sticks.
The length of a lollipop stick produced by the machine is normally distributed with unknown mean \(\mu\) and standard deviation 0.2

Farhan believes that the machine is not working properly and the mean length of the lollipop sticks has decreased.
He takes a random sample of size \(n\) to test, at the 1% level of significance, the hypotheses

Given that the actual value of \(\mu\) is 14.9

5. Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red.
Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.

The probability of Asha getting a red counter on any one draw is 0.07

The probability of Davinda getting a red counter on any one draw is \(p\).
Davinda draws counters until she gets \(n\) red counters.  The random variable \(D\) is the number of counters Davinda draws.

Given that the mean and the standard deviation of \(D\) are 4400 and 660 respectively,

Jerry believes that his bag contains a smaller proportion of red counters than Asha’s bag. To test his belief, Jerry draws counters from his bag until he gets a red counter.  Jerry defines the random variable \(J\) to be the number of counters drawn up to and including the first red counter.

Jerry gets a red counter for the first time on his 34th draw.

Given that the probability of Jerry getting a red counter on any one draw is 0.011

7. A six-sided die has sides labelled 1, 2, 3, 4, 5 and 6

The random variable \(S\) represents the score when the die is rolled.

Alicia rolls the die 45 times and the mean score, \(\overline{S}\), is calculated.

Assuming the die is fair and using a suitable approximation,

Alicia considers the following hypotheses:

\(\mathrm{H}_0\): The die is fair

\(\mathrm{H}_1\): The die is not fair

If \(\overline{S} \lt 3.1\) or \(\overline{S} \gt 3.9\), then \(\mathrm{H}_0\) will be rejected.

Given that the true distribution of \(S\) has mean 4 and variance 3

1. The number of customers entering Jeff’s supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.

Using this information,

A rival supermarket is opened nearby.  Following its opening, the number of customers entering Jeff’s supermarket over a randomly selected 40-minute period is found to be 10

A further randomly selected 20-minute period is observed and the hypothesis test is repeated.
Given that the true rate of customers entering Jeff’s supermarket is now 1 every 5 minutes,

5. 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.

Data from the series of 3-month periods are recorded for 2 years.

Given that the number of accidents per month on the bypass, after the work is completed, is actually 2.1 per month,

5. A machine makes posts. The length of a post is normally distributed with unknown mean \(\mu\) and standard deviation 4 cm.

A random sample of size \(n\) is taken to test, at the 5% significance level, the hypotheses

The manufacturer requires the probability of a Type II error to be less than 0.1 when the actual value of \(\mu\) is 152

2. The number of accidents per year in Daftstown follows a Poisson distribution with mean \(\lambda\). The value of \(\lambda\) has previously been 6 but Jonty claims that since the Council increased the speed limit, the value of \(\lambda\) has increased.

Jonty records the number of accidents in Daftstown in the first year after the speed limit was increased. He plans to test, at the 5% significance level, whether or not there is evidence of an increase in the mean number of accidents in Daftstown per year.

Given that there were 9 accidents in the first year after the speed limit was increased,

4. A manufacturer produces boxes of screws containing short screws and long screws. The manufacturer claims that the probability, \(p\), of a randomly selected screw being long, is 0.5

A shopkeeper does not believe the manufacturer’s claim. He designs two tests, \(A\) and \(B\), to test the hypotheses \(\mathrm{H}_0 : p = 0.5\) and \(\mathrm{H}_1 : p \lt 0.5\)

In test \(A\), a random sample of 10 screws is taken from a box of screws and \(\mathrm{H}_0\) is rejected if there are fewer than 3 long screws.

In test \(B\), a random sample of 5 screws is taken from a box of screws and \(\mathrm{H}_0\) is rejected if there are no long screws, otherwise a second random sample of 5 screws is taken from a box of screws. If there are no long screws in this second sample \(\mathrm{H}_0\) is rejected, otherwise it is accepted.

Some values, to 2 decimal places, of the power function for test \(A\) and the power function for test \(B\) are given in the table below.

The shopkeeper believes that the value of \(p\) is less than 0.4

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.

Given that there are 3 times as many sweets with soft centres as there are sweets with hard centres,

4. A poultry farm produces eggs which are sold in boxes of 6. The farmer believes that the proportion, \(p\), of eggs that are cracked when they are packed in the boxes is approximately 5%. She decides to test the hypotheses

To test these hypotheses she randomly selects a box of eggs and rejects \(\mathrm{H}_0\) if the box contains 2 or more eggs that are cracked. If the box contains 1 egg that is cracked, she randomly selects a second box of eggs and rejects \(\mathrm{H}_0\) if it contains at least 1 egg that is cracked. If the first or the second box contains no cracked eggs, \(\mathrm{H}_0\) is immediately accepted and no further boxes are sampled.

Given that \(p = 0.1\)

Given that \(p = 0.1\) is an unacceptably high value for the farmer,

2. The cloth produced by a certain manufacturer has defects that occur randomly at a constant rate of \(\lambda\) per square metre. If \(\lambda\) is thought to be greater than 1.5 then action has to be taken.

Using \(\mathrm{H}_0 : \lambda = 1.5\) and \(\mathrm{H}_1 : \lambda \gt 1.5\) a quality control officer takes a 4 m\(^2\) sample of cloth and rejects \(\mathrm{H}_0\) if there are 11 or more defects. If there are 8 or fewer defects she accepts \(\mathrm{H}_0\). If there are 9 or 10 defects a second sample of 4 m\(^2\) is taken and \(\mathrm{H}_0\) is rejected if there are 11 or more defects in this second sample, otherwise it is accepted.

5. A statistician believes a coin is biased and the probability, \(p\), of getting a head when the coin is tossed is less than 0.5

The statistician decides to test this by tossing the coin 10 times and recording the number, \(X\), of heads. He sets up the hypotheses \(\mathrm{H}_0 : p = 0.5\) and \(\mathrm{H}_1 : p \lt 0.5\) and rejects the null hypothesis if \(x \lt 3\)

Table 1 gives values, to 2 decimal places, of the power function for the statistician’s test.

Table 1

2.

Rolls of material, manufactured by a machine, contain defects at a mean rate of 6 per roll.

The machine is modified. A single roll is selected at random and a test is carried out to see whether or not the mean number of defects per roll has decreased. The significance level is chosen to be as close as possible to 5%.

7. A machine produces bricks. The lengths, \(x\) mm, of the bricks are distributed \(\mathrm{N}(\mu, 2^2)\).
At the start of each week a random sample of \(n\) bricks is taken to check the machine is working correctly.
A test is then carried out at the 1% level of significance with

The probability of a type II error, when \(\mu = 200\), is less than 0.05

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 \(\mathrm{H}_1 : \lambda \gt 2\), where \(\lambda\) is the mean number of houses sold per week, and rejects the null hypothesis if \(x \geqslant 3\)

The table below gives the values of the power function to 2 decimal places.

Table 1

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.

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.

Table 1 below gives some values, to 2 decimal places, of the power function of the statistician’s test.

Table 1

Figure 1 shows a graph of the power function for the scientist’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

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 \(\mathrm{H}_1 : p \gt 0.05\), 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.

Table 1 below gives some values, to 2 decimal places, of the power function of this test.

Table 1

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.

Figure 1 shows the graph of the power function of the test used by the consultant.

3. A manager in a sweet factory believes that the machines are working incorrectly and the proportion \(p\) of underweight bags of sweets is more than 5%. He decides to test this by randomly selecting a sample of 5 bags and recording the number \(X\) that are underweight. The manager sets up the hypotheses \(\mathrm{H}_0 : p = 0.05\) and \(\mathrm{H}_1 : p \gt 0.05\) and rejects the null hypothesis if \(x \gt 1\).

The manager goes on holiday and his deputy checks the production by randomly selecting a sample of 10 bags of sweets. He rejects the hypothesis that \(p = 0.05\) if more than 2 underweight bags are found in the sample.

The table below gives some values, to 2 decimal places, of the power function for the deputy’s test.

The graph of the power function for the manager’s test is shown in Figure 1.

The deputy suggests that they should use his sampling method rather than the manager’s.

3. Define, in terms of \(\mathrm{H}_0\) and/or \(\mathrm{H}_1\),

The probability of getting a head when a coin is tossed is denoted by \(p\).

This coin is tossed 12 times in order to test the hypotheses \(\mathrm{H}_0 : p = 0.5\) against \(\mathrm{H}_1 : p \neq 0.5\), using a 5% level of significance.

6. A drug is claimed to produce a cure to a certain disease in 35% of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.

The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of \(p\) where \(p\) is the probability of the drug curing a person with the disease.

6. A butter packing machine cuts butter into blocks. The weight of a block of butter is normally distributed with a mean weight of 250 g and a standard deviation of 4 g. A random sample of 15 blocks is taken to monitor any change in the mean weight of the blocks of butter.

5. The number of tornadoes per year to hit a particular town follows a Poisson distribution with mean \(\lambda\). A weatherman claims that due to climate changes the mean number of tornadoes per year has decreased. He records the number of tornadoes \(x\) to hit the town last year.

To test the hypotheses \(\mathrm{H}_0 : \lambda = 7\) and \(\mathrm{H}_1 : \lambda \lt 7\), a critical region of \(x \leqslant 3\) is used.

5. Rolls of cloth delivered to a factory contain defects at an average rate of \(\lambda\) per metre. A quality assurance manager selects a random sample of 15 metres of cloth from each delivery to test whether or not there is evidence that \(\lambda \gt 0.3\). The criterion that the manager uses for rejecting the hypothesis that \(\lambda = 0.3\) is that there are 9 or more defects in the sample.

Table 1 gives some values, to 2 decimal places, of the power function of this test.

Table 1

The manager would like to design a test, of whether or not \(\lambda \gt 0.3\), that uses a smaller length of cloth. He chooses a length of 10 m and requires the probability of a type I error to be less than 10%.

Table 2 gives some values, to 2 decimal places, of the power function for the test in part (c).

Table 2

The cost of wrongly rejecting a delivery of cloth with \(\lambda = 0.3\) is low. Deliveries of cloth with \(\lambda \gt 0.7\) are unusual.

4. The number of accidents that occur at a crossroads has a mean of 3 per month. In order to improve the flow of traffic the priority given to traffic is changed. Colin believes that since the change in priority the number of accidents has increased. He tests his belief by recording the number of accidents \(x\) in the month following the change. Colin sets up the hypotheses \(\mathrm{H}_0 : \lambda = 3\) and \(\mathrm{H}_1 : \lambda \gt 3\), where \(\lambda\) is the mean number of accidents per month, and rejects the null hypothesis if \(x \gt 4\).

The table gives the values of the power function of the test to two decimal places.

2.

A manufacturer sells socks in boxes of 50.

The mean number of faulty socks per box is 7.5. In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.

5. Define

Jane claims that she can read Alan’s mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan’s mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times \(X\) that Jane is correct is recorded.

The probability of Jane stating the correct symbol is denoted by \(p\).

To test the hypothesis \(\mathrm{H}_0 : p = 0.25\) against \(\mathrm{H}_1 : p \gt 0.25\), a critical region of \(X \gt 6\) is used.

Given that \(p = 0.3\), calculate