5.05c Hypothesis test: normal distribution for population mean

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 engineering firm buys steel rods. The steel rods from its present supplier are known to have a mean tensile strength of 230 N/mm\(^2\). A new supplier of steel rods offers to supply rods at a cheaper price than the present supplier. A random sample of ten rods from this new supplier gave tensile strengths, \(x\) N/mm\(^2\), which are summarised below.

Sample size	\(\Sigma x\)	\(\Sigma x^2\)
10	2283	524079

1. Stating your hypotheses clearly, and using a 5\% level of significance, test whether or not the rods from the new supplier have a tensile strength lower than the present supplier. (You may assume that the tensile strength is normally distributed). [7]
2. In the light of your conclusion to part (a) write down what you would recommend the engineering firm to do. [1]

A company manufactures bolts with a mean diameter of 5 mm. The company wishes to check that the diameter of the bolts has not decreased. A random sample of 10 bolts is taken and the diameters, \(x\) mm, of the bolts are measured. The results are summarised below. $$\sum x = 49.1 \quad \sum x^2 = 241.2$$ Using a 1\% level of significance, test whether or not the mean diameter of the bolts is less than 5 mm. (You may assume that the diameter of the bolts follows a normal distribution.) [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]

A teacher wishes to test whether playing background music enables students to complete a task more quickly. The same task was completed by 15 students, divided at random into two groups. The first group had background music playing during the task and the second group had no background music playing. The times taken, in minutes, to complete the task are summarised below.

	Sample size \(n\)	Standard deviation \(s\)	Mean \(\bar{x}\)
With background music	8	4.1	15.9
Without background music	7	5.2	17.9

You may assume that the times taken to complete the task by the students are two independent random samples from normal distributions.

1. Stating your hypotheses clearly, test, at the 10\% level of significance, whether or not the variances of the times taken to complete the task with and without background music are equal. [5]
2. Find a 99\% confidence interval for the difference in the mean times taken to complete the task with and without background music. [7]

Experiments like this are often performed using the same people in each group.

1. Explain why this would not be appropriate in this case. [1]

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]

A car manufacturer claims that, on a motorway, the mean number of miles per gallon for the Panther car is more than 70. To test this claim a car magazine measures the number of miles per gallon, \(x\), of each of a random sample of 20 Panther cars and obtained the following statistics. $$\bar{x} = 71.2 \quad s = 3.4$$ The number of miles per gallon may be assumed to be normally distributed.

1. Stating your hypotheses clearly and using a 5\% level of significance, test the manufacturer's claim. [5]

The standard deviation of the number of miles per gallon for the Tiger car is 4.

1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not there is evidence that the variance of the number of miles per gallon for the Panther car is different from that of the Tiger car. [6]

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]

The weights of the contents of breakfast cereal boxes are normally distributed. A manufacturer changes the style of the boxes but claims that the weight of the contents remains the same. A random sample of 6 old style boxes had contents with the following weights (in grams). 512, 503, 514, 506, 509, 515 The weights, \(y\) grams, of the contents of an independent random sample of 5 new style boxes gave $$\bar{y} = 504.8 \text{ and } s_y = 3.420$$

1. Use a two-tail test to show, at the 10\% level of significance, that the variances of the weights of the contents of the old and new style boxes can be assumed to be equal. State your hypotheses clearly. [5]
2. Showing your working clearly, find a 90\% confidence interval for \(\mu_x - \mu_y\) where \(\mu_x\) and \(\mu_y\) are the mean weights of the contents of old and new style boxes respectively. [7]
3. With reference to your confidence interval comment on the manufacturer's claim. [2]

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 recent census in the U.K. revealed that the heights of females in the U.K. have a mean of 160.9 cm. A doctor is studying the heights of female Indians in a remote region of South America. The doctor measured the height, \(x\) cm, of each of a random sample of 30 female Indians and obtained the following statistics. $$\Sigma x = 4400.7, \quad \Sigma x^2 = 646904.41.$$ The heights of female Indians may be assumed to follow a normal distribution. The doctor presented the results of the study in a medical journal and wrote 'the female Indians in this region are more than 10 cm shorter than females in the U.K.'

1. Stating your hypotheses clearly and using a 5% level of significance, test the doctor's statement. [6]

The census also revealed that the standard deviation of the heights of U.K. females was 6.0 cm.

1. Stating your hypotheses clearly test, at the 5% level of significance, whether or not there is evidence that the variance of the heights of female Indians is different from that of females in the U.K. [6]

The times, \(x\) seconds, taken by the competitors in the 100 m freestyle events at a school swimming gala are recorded. The following statistics are obtained from the data.

	No. of competitors	Sample Mean \(\overline{x}\)	\(\Sigma x^2\)
Girls	8	83.10	55746
Boys	7	88.90	56130

Following the gala a proud parent claims that girls are faster swimmers than boys. Assuming that the times taken by the competitors are two independent random samples from normal distributions,

1. test, at the 10% level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly. [7]
2. Stating your hypotheses clearly, test the parent's claim. Use a 5% level of significance. [6]

A beach is divided into two areas \(A\) and \(B\). A random sample of pebbles is taken from each of the two areas and the length of each pebble is measured. A sample of size 26 is taken from area \(A\) and the unbiased estimate for the population variance is \(s_A^2 = 0.495\) mm\(^2\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s_B^2 = 1.04\) mm\(^2\).

1. Stating your hypotheses clearly test, at the 10\% significance level, whether or not there is a difference in variability of pebble length between area \(A\) and area \(B\). [5]
2. State the assumption you have made about the populations of pebble lengths in order to carry out the test. [1]

1. Define
   1. a Type I error,
   2. a Type II error. [2]

A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.

1. Assuming that the number of chicks reared per year follows a Poisson distribution test, at the 5\% significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly. [4]
2. Calculate the probability of the Type I error for this test. [3]
3. Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error. [2]

Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml, are given in the table. One statistician suggests performing a two-sample \(t\)-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.

1. Stating your hypotheses clearly and using a 5\% significance level, carry out this test. (You may assume \(\bar{x}_A = 26.125\), \(s_A^2 = 7.84\), \(\bar{x}_B = 25\), \(s_B^2 = 4\) and \(\sigma_A^2 = \sigma_B^2\)) [7]

Another statistician suggests analysing these data using a paired \(t\)-test.

1. Using a 5\% significance level, carry out this test. [9]
2. State which of these two tests you consider to be more appropriate. Give a reason for your choice. [1]

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]

A biologist investigating the shell size of turtles takes random samples of adult female and adult male turtles and records the length, \(x\) cm, of the shell. The results are summarised below.

	Number in sample	Sample mean \(\bar{x}\)	\(\sum x^2\)
Female	6	19.6	2308.01
Male	12	13.7	2262.57

You may assume that the samples come from independent normal distributions with the same variance. The biologist claims that the mean shell length of adult female turtles is 5 cm longer than the mean shell length of adult male turtles.

1. Test the biologist's claim at the 5\% level of significance. [10]
2. Given that the true values for the variance of the population of adult male turtles and adult female turtles are both 0.9 cm\(^2\),
   1. show that when samples of size 6 and 12 are used with a 5\% level of significance, the biologist's claim will be accepted if \(4.07 < \bar{X}_F - \bar{X}_M < 5.93\) where \(\bar{X}_F\) and \(\bar{X}_M\) are the mean shell lengths of females and males respectively.
   2. Hence find the probability of a type II error for this test if in fact the true mean shell length of adult female turtles is 6 cm more than the mean shell length of adult male turtles. [6]

The sample variance of the lengths of a random sample of 9 paving slabs sold by a builders' merchant is 36 mm\(^2\). The sample variance of the lengths of a random sample of 11 paving slabs sold by a second builders' merchant is 225 mm\(^2\). Test at the 10\% significance level whether or not there is evidence that the lengths of paving slabs sold by these builders' merchants differ in variability. State your hypotheses clearly. (You may assume the lengths of paving slabs are normally distributed.) [5]

A newspaper runs a daily Sudoku. A random sample of 10 people took the following times, in minutes, to complete the Sudoku. 5.0 \quad 4.5 \quad 4.7 \quad 5.3 \quad 5.2 \quad 4.1 \quad 5.3 \quad 4.8 \quad 5.5 \quad 4.6 Given that the times to complete the Sudoku follow a normal distribution,

1. calculate a 95\% confidence interval for
   1. the mean,
   2. the variance,
   of the times taken by people to complete the Sudoku. [13] The newspaper requires the average time needed to complete the Sudoku to be 5 minutes with a standard deviation of 0.7 minutes.
2. Comment on whether or not the Sudoku meets this requirement. Give a reason for your answer. [3]

Boxes of chocolates manufactured by Philippe have a mean weight of \(\mu\) grams and a standard deviation of \(\sigma\) grams. A random sample of 25 of these boxes are weighed. Using this sample, the unbiased estimate of \(\mu\) is 455 and the unbiased estimate of \(\sigma^2\) is 55.

1. Test, at the 5\% level of significance, whether or not \(\sigma\) is greater than 6. State your hypotheses clearly. [6]
2. Test, at the 5\% level of significance, whether or not \(\mu\) is more than 450. [6]
3. State an assumption you have made in order to carry out the above tests. [1]

A survey during 2013 investigated mean expenditure on bread and on alcohol. The 2013 survey obtained information from 12 144 adults. The survey revealed that the mean expenditure per adult per week on bread was 127p.

1. For 2012, it is known that the expenditure per adult per week on bread had mean 123p, and a standard deviation of 70p.
   1. Carry out a hypothesis test, at the 5% significance level, to investigate whether the mean expenditure per adult per week on bread changed from 2012 to 2013. Assume that the survey data is a random sample taken from a normal distribution. [5 marks]
   2. Calculate the greatest and least values for the sample mean expenditure on bread per adult per week for 2013 that would have resulted in acceptance of the null hypothesis for the test you carried out in part (a)(i). Give your answers to two decimal places. [2 marks]
2. The 2013 survey revealed that the mean expenditure per adult, per week on alcohol was 324p. The mean expenditure per adult per week on alcohol for 2009 was 307p. A test was carried out on the following hypotheses relating to mean expenditure per adult per week on alcohol in 2013. \(H_0 : \mu = 307\) \(H_1 : \mu \neq 307\) This test resulted in the null hypothesis, \(H_0\), being rejected. State, with a reason, whether the test result supports the following statements:
   1. the mean UK expenditure on alcohol per adult per week increased by 17p from 2009 to 2013; [2 marks]
   2. the mean UK consumption of alcohol per adult per week changed from 2009 to 2013. [2 marks]

A company produces kettlebells whose weights are normally distributed with mean \(16\) kg and standard deviation \(0.08\) kg.

1. Find the probability that the weight of a randomly selected kettlebell is greater than \(16.05\) kg. [2]

The company trials a new production method. It needs to check that the mean is still \(16\) kg. It assumes that the standard deviation is unchanged. The company takes a random sample of 25 kettlebells and it decides to reject the new production method if the sample mean does not round to \(16\) kg to the nearest \(100\) g.

1. Find the probability that the new production method will be rejected if, in fact, the mean is still \(16\) kg. [4]

The company decides instead to use a 5\% significance test. A random sample of 25 kettlebells is selected and the mean is found to be \(16.02\) kg.

1. Carry out the test to determine whether or not the new production method will be rejected. [6]

A manufacturer of batteries for electric cars claims that an hour of charge can power a certain model of car to travel for an average of 123 miles. An electric car company and a consumer, Hopcyn, both wish to test the validity of the manufacturer's claim.

1. Explain why Hopcyn may want to use a one-sided test and why the car company may want to use a two-sided test. [2]

To test the validity of this claim, Hopcyn collects data from a random sample of 90 drivers of this model of car to see how far they travelled, \(X\) miles, on an hour of charge. He produced the following summary statistics. $$\sum x = 11007 \quad \sum x^2 = 1361913$$

1. 1. Assuming Hopcyn uses a one-sided test, state the hypotheses.
   2. Test at the 5\% significance level whether the manufacturer's claim is correct. [8]

The sports performance director at a university wishes to investigate whether there is a difference in the means of the specific gravities of blood of cyclists and runners. She models the distribution of specific gravity for cyclists as \(N\left(\mu_c, 8^2\right)\) and for runners as \(N\left(\mu_r, 10^2\right)\).

1. State suitable hypotheses for this investigation. [1]

The mean specific gravity of blood of a random sample of 40 cyclists from the university was 1063. The mean specific gravity of blood of a random sample of 40 runners from the same university was 1060.

1. Calculate and interpret the \(p\)-value for the data. [6]
2. Suppose now that both samples were of size \(n\), instead of 40. Find the least value of \(n\) that would ensure that an observed difference of 3 in the mean specific gravities would be significant at the 1\% level. [4]

A factory manufactures a certain type of string. In order to ensure the quality of the product, a random sample of 10 pieces of string is taken every morning and the breaking strength of each piece, in Newtons, is measured. One morning, the results are as follows. $$68.1 \quad 70.4 \quad 68.6 \quad 67.7 \quad 71.3 \quad 67.6 \quad 68.9 \quad 70.2 \quad 68.4 \quad 69.8$$ You may assume that this is a random sample from a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma^2\).

1. Determine a 95% confidence interval for \(\mu\). [9]
2. The factory manager is given these results and he asks 'Can I assume that the confidence interval that you have given me contains \(\mu\) with probability 0.95?' Explain why the answer to this question is no and give a correct interpretation. [2]