5.05c Hypothesis test: normal distribution for population mean

80 randomly chosen people are asked to estimate a time interval of 60 seconds without using a watch or clock. The mean of the 80 estimates is 58.9 seconds. Previous evidence shows that the population standard deviation of such estimates is 5.0 seconds. Test, at the 5% significance level, whether there is evidence that people tend to underestimate the time interval. [7]

It is desired to test whether the average amount of sleep obtained by school pupils in Year 11 is 8 hours, based on a random sample of size 64. The population standard deviation is 0.87 hours and the sample mean is denoted by \(\bar{H}\). The critical values for the test are \(\bar{H} = 7.72\) and \(\bar{H} = 8.28\).

1. State appropriate hypotheses for the test, explaining the meaning of any symbol you use. [3]
2. Calculate the significance level of the test. [4]
3. Explain what is meant by a Type I error in this context. [1]
4. Given that in fact the average amount of sleep obtained by all pupils in Year 11 is 7.9 hours, find the probability that the test results in a Type II error. [3]

It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years\(^2\).

1. Test at the 5% significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild. [7]
2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years\(^2\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test. [5]
3. Explain whether the Central Limit Theorem is needed in these tests. [1]

The manufacturer's specification for batteries used in a certain electronic game is that the mean lifetime should be 32 hours. The manufacturer tests a random sample of 10 batteries made in Factory A, and the lifetimes (\(x\) hours) are summarised by \(n = 10\), \(\sum x = 289.0\) and \(\sum x^2 = 8586.19\). It may be assumed that the population of lifetimes has a normal distribution.

1. Carry out a one-tail test at the \(5\%\) significance level of whether the specification is being met. [7]
2. Justify the use of a one-tail test in this context. [1]

Batteries made with the same specification are also made in Factory B. The lifetimes of these batteries are also normally distributed. A random sample of 12 batteries from this factory was tested. The lifetimes are summarised by \(n = 12\), \(\sum x = 363.0\) and \(\sum x^2 = 11290.95\).

1. 1. State what further assumption must be made in order to test whether there is any difference in the mean lifetimes of batteries made at the two factories. Use the data to comment on whether this assumption is reasonable. [3]
   2. Carry out the test at the \(10\%\) significance level. [7]

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]

William Sealy, a biochemistry student, is doing work experience at a brewery. One of his tasks is to monitor the specific gravity of the brewing mixture during the brewing process. For one particular recipe, an initial specific gravity of 1.040 is required. A random sample of 9 measurements of the specific gravity at the start of the process gave the following results. 1.046 \quad 1.048 \quad 1.039 \quad 1.055 \quad 1.038 \quad 1.054 \quad 1.038 \quad 1.051 \quad 1.038

1. William has to test whether the specific gravity of the mixture meets the requirement. Why might a \(t\) test be used for these data and what assumption must be made? [3]
2. Carry out the test using a significance level of 10\%. [9]
3. Find a 95\% confidence interval for the true mean specific gravity of the mixture and explain what is meant by a 95\% confidence interval. [6]

A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet. The data on the height, \(V\) cm, of the 80 boys who had always eaten a vegetarian diet is summarised by $$\Sigma V = 10\,367, \quad \Sigma V^2 = 1\,350\,314.$$

1. Calculate unbiased estimates of the mean and variance of \(V\). [5]

The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and 96.24 cm\(^2\) respectively.

1. Stating your hypotheses clearly, test at the 1% level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet. [8]

For a project, a student is investigating whether more athletic individuals have better hand-eye coordination. He records the time it takes a number of students to complete a task testing coordination skills and notes whether or not they play for a school sports team. His results are as follows: Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence that those who play in a school team complete the task more quickly on average. [8 marks]

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 \(s_A^2 = 0.495 \text{ mm}^2\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s_B^2 = 1.04 \text{ mm}^2\).

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 this test. [1]

A random sample of 10 mustard plants had the following heights, in mm, after 4 days growth. 5.0, 4.5, 4.8, 5.2, 4.3, 5.1, 5.2, 4.9, 5.1, 5.0 Those grown previously had a mean height of 5.1 mm after 4 days. Using a 2.5\% significance level, test whether or not the mean height of these plants is less than that of those grown previously. (You may assume that the height of mustard plants after 4 days follows a normal distribution.) [9]

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. \includegraphics{figure_7} 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. [(b)] 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 mechanic is required to change car tyres. An inspector timed a random sample of 20 tyre changes and calculated the unbiased estimate of the population variance to be 6.25 minutes². Test, at the 5\% significance level, whether or not the standard deviation of the population of times taken by the mechanic is greater than 2 minutes. State your hypotheses clearly. [6]

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 supervisor wishes to check the typing speed of a new typist. On 10 randomly selected occasions, the supervisor records the time taken for the new typist to type 100 words. The results, in seconds, are given below. 110, 125, 130, 126, 128, 127, 118, 120, 122, 125 The supervisor assumes that the time taken to type 100 words is normally distributed.

1. Calculate a 95\% confidence interval for
   1. the mean,
   2. the variance
   of the population of times taken by this typist to type 100 words. [13]

The supervisor requires the average time needed to type 100 words to be no more than 130 seconds and the standard deviation to be no more than 4 seconds.

1. [(b)] Comment on whether or not the supervisor should be concerned about the speed of the new typist. [3]

A grocer receives deliveries of cauliflowers from two different growers, \(A\) and \(B\). The grocer takes random samples of cauliflowers from those supplied by each grower. He measures the weight \(x\), in grams, of each cauliflower. The results are summarised in the table below. \includegraphics{figure_7}

1. Show, at the 10\% significance level, that the variances of the populations from which the samples are drawn can be assumed to be equal by testing the hypothesis H₀: \(\sigma_A^2 = \sigma_B^2\) against hypothesis H₁: \(\sigma_A^2 \neq \sigma_B^2\). (You may assume that the two samples come from normal populations.) [6]

The grocer believes that the mean weight of cauliflowers provided by \(B\) is at least 150 g more than the mean weight of cauliflowers provided by \(A\).

1. [(b)] Use a 5\% significance level to test the grocer's belief. [8]
2. Justify your choice of test. [2]

The standard deviation of the length of a random sample of 8 fence posts produced by a timber yard was 8 mm. A second timber yard produced a random sample of 13 fence posts with a standard deviation of 14 mm.

1. Test, at the 10\% significance level, whether or not there is evidence that the lengths of fence posts produced by these timber yards differ in variability. State your hypotheses clearly. [5]
2. State an assumption you have made in order to carry out the test in part (a). [1]

(Total 6 marks)

A machine is set to fill bags with flour such that the mean weight is 1010 grams. To check that the machine is working properly, a random sample of 8 bags is selected. The weight of flour, in grams, in each bag is as follows. 1010 1015 1005 1000 998 1008 1012 1007 Carry out a suitable test, at the 5\% significance level, to test whether or not the mean weight of flour in the bags is less than 1010 grams. (You may assume that the weight of flour delivered by the machine is normally distributed.) (Total 8 marks)

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)

Historical records from a large colony of squirrels show that the weight of squirrels is normally distributed with a mean of 101.2 g. Following a change in the diet of squirrels, a biologist is interested in whether or not the mean weight has changed. A random sample of 14 squirrels is weighed and their weights \(x\), in grams, recorded. The results are summarised as follows: \(\sum x = 1370\), \(\sum x^2 = 134487.50\). Stating your hypotheses clearly test, at the 5\% level of significance, whether or not there has been a change in the mean weight of the squirrels. [7]

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 lengths, \(x\) mm, of the forewings of a random sample of male and female adult butterflies are measured. The following statistics are obtained from the data. \includegraphics{figure_3}

1. Assuming the lengths of the forewings are normally distributed 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, at the 5\% level of significance, whether the mean length of the forewings of the female butterflies is less than the mean length of the forewings of the male butterflies. [6]

A large number of students are split into two groups \(A\) and \(B\). The students sit the same test but under different conditions. Group A has music playing in the room during the test, and group B has no music playing during the test. Small samples are then taken from each group and their marks recorded. The marks are normally distributed. The marks are as follows: Sample from Group \(A\): 42, 40, 35, 37, 34, 43, 42, 44, 49 Sample from Group \(B\): 40, 44, 38, 47, 38, 37, 33

1. Stating your hypotheses clearly, and using a 10\% level of significance, test whether or not there is evidence of a difference between the variances of the marks of the two groups. [8]
2. State clearly an assumption you have made to enable you to carry out the test in part (a). [1]
3. Use a two tailed test, with a 5\% level of significance, to determine if the playing of music during the test has made any difference in the mean marks of the two groups. State your hypotheses clearly. [7]
4. Write down what you can conclude about the effect of music on a student's performance during the test. [1]