Paired sample t-test

A production line has two machines, A and B, for delivering liquid soap into bottles. Each machine is set to deliver a nominal amount of 250 ml, but it is not expected that they will work to a high level of accuracy. In particular, it is known that the ambient temperature affects the rate of flow of the liquid and leads to variation in the amounts delivered. The operators think that machine B tends to deliver a somewhat greater amount than machine A, no matter what the ambient temperature. This is being investigated by an experiment. A random sample of 10 results from the experiment is shown below. Each column of data is for a different ambient temperature.

Ambient temperature	\(T_1\)	\(T_2\)	\(T_3\)	\(T_4\)	\(T_5\)	\(T_6\)	\(T_7\)	\(T_8\)	\(T_9\)	\(T_{10}\)
Amount delivered by machine A	246.2	251.6	252.0	246.6	258.4	251.0	247.5	247.1	248.1	253.4
Amount delivered by machine B	248.3	252.6	252.8	247.2	258.8	250.0	247.2	247.9	249.0	254.5

1. Use an appropriate \(t\) test to examine, at the 5\% level of significance, whether the mean amount delivered by machine B may be taken as being greater than that delivered by machine A, stating carefully your null and alternative hypotheses and the required distributional assumption. [11]
2. Using the data for machine A in the table above, provide a two-sided 95\% confidence interval for the mean amount delivered by this machine, stating the required distributional assumption. Explain whether you would conclude that the machine appears to be working correctly in terms of the nominal amount as set. [7]

A doctor believes that the span of a person's dominant hand is greater than that of the weaker hand. To test his theory, the doctor measures the spans of the dominant and weaker hands of a random sample of 8 people. He subtracts the span of the weaker hand from that of the dominant hand. The spans, in cm, are summarised in the table below. \includegraphics{figure_4} Test, at the 5\% significance level, the doctor's belief. [9]

A farmer set up a trial to assess the effect of two different diets on the increase in the weight of his lambs. He randomly selected 20 lambs. Ten of the lambs were given diet \(A\) and the other 10 lambs were given diet \(B\). The gain in weight, in kg, of each lamb over the period of the trial was recorded.

1. State why a paired \(t\)-test is not suitable for use with these data. [1]
2. Suggest an alternative method for selecting the sample which would make the use of a paired \(t\)-test suitable. [1]
3. Suggest two other factors that the farmer might consider when selecting the sample. [2]

The following paired data were collected. \includegraphics{figure_4}

1. [(d)] Using a paired \(t\)-test, at the 5\% significance level, test whether or not there is evidence of a difference in the weight gained by the lambs using diet \(A\) compared with those using diet \(B\). [8]
2. State, giving a reason, which diet you would recommend the farmer to use for his lambs. [1]

(Total 13 marks)

As part of an investigation into the effectiveness of solar heating, a pair of houses was identified where the mean weekly fuel consumption was the same. One of the houses was then fitted with solar heating and the other was not. Following the fitting of the solar heating, a random sample of 9 weeks was taken and the table below shows the weekly fuel consumption for each house. \includegraphics{figure_3}

1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not there is evidence that the solar heating reduces the mean weekly fuel consumption. [8]
2. State an assumption about weekly fuel consumption that is required to carry out this test. [1]

A medical student is investigating two methods of taking a person's blood pressure. He takes a random sample of 10 people and measures their blood pressure using an arm cuff and a finger monitor. The table below shows the blood pressure for each person, measured by each method. \includegraphics{figure_1}

1. Use a paired \(t\)-test to determine, at the 10\% level of significance, whether or not there is a difference in the mean blood pressure measured using the two methods. State your hypotheses clearly. [8]
2. State an assumption about the underlying distribution of measured blood pressure required for this test. [1]

The weights, in grams, of mice are normally distributed. A biologist takes a random sample of 10 mice. She weighs each mouse and records its weight. The ten mice are then fed on a special diet. They are weighed again after two weeks. Their weights in grams are as follows:

Mouse	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Weight before diet	50.0	48.3	47.5	54.0	38.9	42.7	50.1	46.8	40.3	41.2
Weight after diet	52.1	47.6	50.1	52.3	42.2	44.3	51.8	48.0	41.9	43.6

Stating your hypotheses clearly, and using a 1\% level of significance, test whether or not the diet causes an increase in the mean weight of the mice. [8]

An emission-control device is tested to see if it reduces CO\(_2\) emissions from cars. The emissions from 6 randomly selected cars are measured with and without the device. The results are as follows.

Car	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)
Emissions without device	151.4	164.3	168.5	148.2	139.4	151.2
Emissions with device	148.9	162.7	166.9	150.1	140.0	146.7

1. State an assumption that needs to be made in order to carry out a \(t\)-test in this case. [1]
2. State why a paired \(t\)-test is suitable for use with these data. [1]
3. Using a 5\% level of significance, test whether or not there is evidence that the device reduces CO\(_2\) emissions from cars. [8]
4. Explain, in context, what a type II error would be in this case. [2]

As part of an investigation, a random sample of 10 people had their heart rate, in beats per minute, measured whilst standing up and whilst lying down. The results are summarised below.

Person	1	2	3	4	5	6	7	8	9	10
Heart rate lying down	66	70	59	65	72	66	62	69	56	68
Heart rate standing up	75	76	63	67	80	75	65	74	63	75

1. State one assumption that needs to be made in order to carry out a paired \(t\)-test. [1]
2. Test, at the 5\% level of significance, whether or not there is any evidence that standing up increases people's mean heart rate by more than 5 beats per minute. State your hypotheses clearly. [8]

Manuel is planning to buy a new machine to squeeze oranges in his cafe and he has two models, at the same price, on trial. The manufacturers of machine B claim that their machine produces more juice from an orange than machine A. To test this claim Manuel takes a random sample of 8 oranges, cuts them in half and puts one half in machine A and the other half in machine B. The amount of juice, in ml, produced by each machine is given in the table below.

Orange	1	2	3	4	5	6	7	8
Machine A	60	58	55	53	52	51	54	56
Machine B	61	60	58	52	55	50	52	58

Stating your hypotheses clearly, test, at the 10\% level of significance, whether or not the mean amount of juice produced by machine B is more than the mean amount produced by machine A. [8]

A chemist has developed a fuel additive and claims that it reduces the fuel consumption of cars. To test this claim, 8 randomly selected cars were each filled with 20 litres of fuel and driven around a race circuit. Each car was tested twice, once with the additive and once without. The distances, in miles, that each car travelled before running out of fuel are given in the table below.

Car	1	2	3	4	5	6	7	8
Distance without additive	163	172	195	170	183	185	161	176
Distance with additive	168	185	187	172	180	189	172	175

Assuming that the distances travelled follow a normal distribution and stating your hypotheses clearly test, at the 10% level of significance, whether or not there is evidence to support the chemist's claim. [8]

A medical student is investigating whether there is a difference in a person's blood pressure when sitting down and after standing up. She takes a random sample of 12 people and measures their blood pressure, in mmHg, when sitting down and after standing up. The results are shown below.

Person	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)	\(K\)	\(L\)
Sitting down	135	146	138	146	141	158	136	135	146	161	119	151
Standing up	131	147	132	140	138	160	127	136	142	154	130	144

The student decides to carry out a paired \(t\)-test to investigate whether, on average, the blood pressure of a person when sitting down is more than their blood pressure after standing up.

1. State clearly the hypotheses that should be used and any necessary assumption that needs to be made. [2]
2. Carry out the test at the 1\% level of significance. [7]