5.05c Hypothesis test: normal distribution for population mean

2 A coffee machine used in a bar is claimed by the manager to dispense 170 ml of coffee per cup on average. A customer believes that the average amount of coffee dispensed is less than 170 ml . She measures the amount of coffee in 6 randomly chosen cups. The results, in ml , are as follows. $$\begin{array} { l l l l l l } 167 & 171 & 164 & 169 & 168 & 166 \end{array}$$ Assuming a relevant normal distribution, test the manager's claim at the 5\% significance level.

6 It has been suggested that people who suffer anxiety when they are about to undergo surgery can have their anxiety reduced by listening to relaxation tapes. A study was carried out on 18 experimental subjects who listened to relaxation tapes, and 13 control subjects who listened to neutral tapes. After listening to the tapes, the subjects were given a test which produced an anxiety score, \(X\). Higher scores indicated higher anxiety. The results are summarised in the table.

1. Use a two-sample \(t\)-test, at the \(5 \%\) significance level, to test whether anxiety is reduced by listening to relaxation tapes. State two necessary assumptions for the validity of your test.
2. State why a test using a normal distribution would not have been appropriate.

2 Four pairs of randomly chosen twins were each given identical puzzles to solve. The times taken (in seconds) are shown in the following table. Stating any necessary assumption, test at the \(10 \%\) significance level whether there is a difference between the population mean times of first-born and second-born twins.

3 A charity raises money by sending letters asking for donations. Because of recent poor responses, the charity's fund-raiser, Anna, decides to alter the letter's appearance and designs two possible alternatives, one colourful and the other plain. She believes that the colourful letter will be more successful. Anna sends 60 colourful letters and 40 plain letters to 100 people randomly chosen from the charity's database. There were 39 positive responses to the colourful letter and 12 positive responses to the plain letter. The population proportions of positive responses to the colourful and plain letters are denoted by \(p _ { C }\) and \(p _ { P }\) respectively. Test the null hypothesis \(p _ { C } - p _ { P } = 0.15\) against the alternative hypothesis \(p _ { C } - p _ { P } > 0.15\) at the \(2 \frac { 1 } { 2 } \%\) significance level and state what Anna could report to her manager.

4 The time interval, \(T\) minutes, between consecutive stoppages of a particular grinding machine is regularly measured. \(T\) is normally distributed with mean \(\mu\).
24 randomly chosen values of \(T\) are summarised by $$\sum _ { i = 1 } ^ { 24 } t _ { i } = 348.0 \text { and } \sum _ { i = 1 } ^ { 24 } t _ { i } ^ { 2 } = 5195.5 .$$

1. Calculate a symmetric \(95 \%\) confidence interval for \(\mu\).
2. For the machine to be working acceptably, \(\mu\) should be at least 15.0 . Using a test at the 10\% significance level, decide whether the machine is working acceptably.

4 A new computer was bought by a local council to search council records and was tested by an employee. She searched a random sample of 500 records and the sample mean search time was found to be 2.18 milliseconds and an unbiased estimate of variance was \(1.58 ^ { 2 }\) milliseconds \({ } ^ { 2 }\).

1. Calculate a \(98 \%\) confidence interval for the population mean search time \(\mu\) milliseconds.
2. It is required to obtain a sample mean time that differs from \(\mu\) by less than 0.05 milliseconds with probability 0.95 . Estimate the sample size required.
3. State why it is unnecessary for the validity of your calculations that search time has a normal distribution.

7 Two machines \(A\) and \(B\) both pack cartons in a factory. The mean packing times are compared by timing the packing of 10 randomly chosen cartons from machine \(A\) and 8 randomly chosen cartons from machine \(B\). The times, \(t\) seconds, taken to pack these cartons are summarised below. The packing times have independent normal distributions.

1. Stating a necessary assumption, carry out a test, at the \(1 \%\) significance level, of whether the population mean packing times differ for the two machines.
2. Find the largest possible value of the constant \(c\) for which there is evidence at the \(1 \%\) significance level that \(\mu _ { B } - \mu _ { A } > c\), where \(\mu _ { B }\) and \(\mu _ { A }\) denote the respective population mean packing times in seconds.

4 A group of students were tested in geography before and after a fieldwork course. The marks of 10 randomly selected students are shown in the table.

1. Use a suitable \(t\)-test, at the \(5 \%\) level of significance, to test whether the students' performance has improved.
2. State the necessary assumption in applying the test.

6 The masses at birth, in kg, of random samples of babies were recorded for each of the years 1970 and 2010. The table shows the sample mean and an unbiased estimate of the population variance for each year.

1. A researcher tests the null hypothesis that babies born in 2010 are 0.04 kg heavier, on average, than babies born in 1970, against the alternative hypothesis that they are more than 0.04 kg heavier on average. Show that, at the \(5 \%\) level of significance, the null hypothesis is not rejected.
2. Another researcher chooses samples of equal size, \(n\), for the two years. Using the same hypothesis as in part (i), she finds that the null hypothesis is rejected at the \(5 \%\) level of significance. Assuming that the sample means and unbiased estimates of population variance for the two years are as given in the table, find the smallest possible value of \(n\).

2 A factory manufactures paperweights consisting of glass mounted on a wooden base. The volume of glass, in \(\mathrm { cm } ^ { 3 }\), in a paperweight has a Normal distribution with mean 56.5 and standard deviation 2.9. The volume of wood, in \(\mathrm { cm } ^ { 3 }\), also has a Normal distribution with mean 38.4 and standard deviation 1.1. These volumes are independent of each other. For the purpose of quality control, paperweights for testing are chosen at random from the factory's output.

1. Find the probability that the volume of glass in a randomly chosen paperweight is less than \(60 \mathrm {~cm} ^ { 3 }\).
2. Find the probability that the total volume of a randomly chosen paperweight is more than \(100 \mathrm {~cm} ^ { 3 }\). The glass has a mass of 3.1 grams per \(\mathrm { cm } ^ { 3 }\) and the wood has a mass of 0.8 grams per \(\mathrm { cm } ^ { 3 }\).
3. Find the probability that the total mass of a randomly chosen paperweight is between 200 and 220 grams.
4. The factory manager introduces some modifications intended to reduce the mean mass of the paperweights to 200 grams or less. The variance is also affected but not the Normality. Subsequently, for a random sample of 10 paperweights, the sample mean mass is 205.6 grams and the sample standard deviation is 8.51 grams. Is there evidence, at the \(5 \%\) level of significance, that the intended reduction of the mean mass has not been achieved?

3 Pathology departments in hospitals routinely analyse blood specimens. Ideally the analysis should be done while the specimens are fresh to avoid any deterioration, but this is not always possible. A researcher decides to study the effect of freezing specimens for later analysis by measuring the concentrations of a particular hormone before and after freezing. He collects and divides a sample of 15 specimens. One half of each specimen is analysed immediately, the other half is frozen and analysed a month later. The concentrations of the particular hormone (in suitable units) are as follows. A \(t\) test is to be used in order to see if, on average, there is a reduction in hormone concentration as a result of being frozen.

1. Explain why a paired test is appropriate in this situation.
2. State the hypotheses that should be used, together with any necessary assumptions.
3. Carry out the test using a \(1 \%\) significance level.
4. A \(p \%\) confidence interval for the true mean reduction in hormone concentration is found to be ( \(0.4869,2.8131\) ). Determine the value of \(p\).

3 Cholesterol is a lipid (fat) which is manufactured by the liver from the fatty foods that we eat. It plays a vital part in allowing the body to function normally. However, when high levels of cholesterol are present in the blood there is a risk of arterial disease. Among the factors believed to assist with achieving and maintaining low cholesterol levels are weight loss and exercise. A doctor wishes to test the effectiveness of exercise in lowering cholesterol levels. For a random sample of 12 of her patients, she measures their cholesterol levels before and after they have followed a programme of exercise. The measurements obtained are as follows.

1. A \(t\) test is to be used in order to see if, on average, the exercise programme seems to be effective in lowering cholesterol levels. State the distributional assumption necessary for the test, and carry out the test using a \(1 \%\) significance level.
2. A second random sample of 12 patients gives a \(95 \%\) confidence interval of \(( - 0.5380,1.4046 )\) for the true mean reduction (before - after) in cholesterol level. Find the mean and standard deviation for this sample. How might the doctor interpret this interval in relation to the exercise programme?

4 The weights of a particular variety (A) of tomato are known to be Normally distributed with mean 80 grams and standard deviation 11 grams.

1. Find the probability that a randomly chosen tomato of variety A weighs less than 90 grams. The weights of another variety (B) of tomato are known to be Normally distributed with mean 70 grams. These tomatoes are packed in sixes using packaging that weighs 15 grams.
2. The probability that a randomly chosen pack of 6 tomatoes of variety B , including packaging, weighs less than 450 grams is 0.8463 . Show that the standard deviation of the weight of single tomatoes of variety B is 6 grams, to the nearest gram.
3. Tomatoes of variety A are packed in fives using packaging that weighs 25 grams. Find the probability that the total weight of a randomly chosen pack of variety A is greater than the total weight of a randomly chosen pack of variety B .
4. A new variety (C) of tomato is introduced. The weights, \(c\) grams, of a random sample of 60 of these tomatoes are measured giving the following results. $$\Sigma c = 3126.0 \quad \Sigma c ^ { 2 } = 164223.96$$ Find a \(95 \%\) confidence interval for the true mean weight of these tomatoes.

1 Each month the amount of electricity, measured in kilowatt-hours ( kWh ), used by a particular household is Normally distributed with mean 406 and standard deviation 12.

1. Find the probability that, in a randomly chosen month, less than 420 kWh is used. The charge for electricity used is 14.6 pence per kWh .
2. Write down the distribution of the total charge for the amount of electricity used in any one month. Hence find the probability that, in a randomly chosen month, the total charge is more than \(\pounds 60\).
3. The household receives a bill every three months. Assume that successive months may be regarded as independent of each other. Find the value of \(b\) such that the probability that a randomly chosen bill is less than \(\pounds b\) is 0.99 . In a different household, the amount of electricity used per month was Normally distributed with mean 432 kWh . This household buys a new washing machine that is claimed to be cheaper to run than the old one. Over the next six months the amounts of electricity used, in kWh , are as follows. $$\begin{array} { l l l l l l } 404 & 433 & 420 & 423 & 413 & 440 \end{array}$$
4. Treating this as a random sample, carry out an appropriate test, with a \(5 \%\) significance level, to see if there is any evidence to suggest that the amount of electricity used per month by this household has decreased on average.

1

1. Define simple random sampling. Describe briefly one difficulty associated with simple random sampling.
2. Freeze-drying is an economically important process used in the production of coffee. It improves the retention of the volatile aroma compounds. In order to maintain the quality of the coffee, technologists need to monitor the drying rate, measured in suitable units, at regular intervals. It is known that, for best results, the mean drying rate should be 70.3 units and anything substantially less than this would be detrimental to the coffee. Recently, a random sample of 12 observations of the drying rate was as follows. $$\begin{array} { l l l l l l l l l l l l } 66.0 & 66.1 & 59.8 & 64.0 & 70.9 & 71.4 & 66.9 & 76.2 & 65.2 & 67.9 & 69.2 & 68.5 \end{array}$$
   1. Carry out a test to investigate at the \(5 \%\) level of significance whether the mean drying rate appears to be less than 70.3. State the distributional assumption that is required for this test.
   2. Find a 95\% confidence interval for the true mean drying rate.

1 A certain industrial process requires a supply of water. It has been found that, for best results, the mean water pressure in suitable units should be 7.8. The water pressure is monitored by taking measurements at regular intervals. On a particular day, a random sample of the measurements is as follows. $$\begin{array} { l l l l l l l l l } 7.50 & 7.64 & 7.68 & 7.51 & 7.70 & 7.85 & 7.34 & 7.72 & 7.74 \end{array}$$ These data are to be used to carry out a hypothesis test concerning the mean water pressure.

1. Why is a test based on the Normal distribution not appropriate in this case?
2. What distributional assumption is needed for a test based on the \(t\) distribution?
3. Carry out a \(t\) test, with a \(2 \%\) level of significance, to see whether it is reasonable to assume that the mean pressure is 7.8 .
4. Explain what is meant by a \(95 \%\) confidence interval.
5. Find a \(95 \%\) confidence interval for the actual mean water pressure.

2 Pat makes and sells fruit cakes at a local market. On her stall a sign states that the average weight of the cakes is 1 kg . A trading standards officer carries out a routine check of a random sample of 8 of Pat's cakes to ensure that they are not underweight, on average. The weights, in kg , that he records are as follows. $$\begin{array} { l l l l l l l l } 0.957 & 1.055 & 0.983 & 0.917 & 1.015 & 0.865 & 1.013 & 0.854 \end{array}$$

1. On behalf of the trading standards officer, carry out a suitable test at a \(5 \%\) level of significance, stating your hypotheses clearly. Assume that the weights of Pat's fruit cakes are Normally distributed.
2. Find a 95\% confidence interval for the true mean weight of Pat's fruit cakes. Pat's husband, Tony, is the owner of a factory which makes and supplies fruit cakes to a large supermarket chain. A large random sample of \(n\) of these cakes has mean weight \(\bar { x } \mathrm {~kg}\) and variance \(0.006 \mathrm {~kg} ^ { 2 }\).
3. Write down, in terms of \(n\) and \(\bar { x }\), a \(95 \%\) confidence interval for the true mean weight of cakes produced in Tony's factory.
4. What is the size of the smallest sample that should be taken if the width of the confidence interval in part (iii) is to be 0.025 kg at most?

1 Gerry runs 5000 -metre races for his local athletics club. His coach has been monitoring his practice times for several months and he believes that they can be modelled using a Normal distribution with mean 15.3 minutes. The coach suggests that Gerry should try running with a pacemaker in order to see if this can improve his times. Subsequently a random sample of 10 of Gerry's times with the pacemaker is collected to see if any reduction has been achieved. The sample of times (in minutes) is as follows. $$\begin{array} { l l l l l l l l l l } 14.86 & 15.00 & 15.62 & 14.44 & 15.27 & 15.64 & 14.58 & 14.30 & 15.08 & 15.08 \end{array}$$

1. Why might a \(t\) test be used for these data?
2. Using a \(5 \%\) significance level, carry out the test to see whether, on average, Gerry's times have been reduced.
3. What is meant by 'a \(5 \%\) significance level'? What would be the consequence of decreasing the significance level?
4. Find a \(95 \%\) confidence interval for the true mean of Gerry's times using a pacemaker.

1 Technologists at a company that manufactures paint are trying to develop a new type of gloss paint with a shorter drying time than the current product. In order to test whether the drying time has been reduced, the technologists paint a square metre of each of the new and old paints on each of 10 different surfaces. The lengths of time, in hours, that each square metre takes to dry are as follows.

1. Explain why a paired sample is used in this context.
2. The mean reduction in drying time is to be investigated. Why might a \(t\) test be appropriate in this context and what assumption needs to be made?
3. Using a significance level of \(5 \%\), carry out a test to see if there appears to be any reduction in mean drying time.
4. Find a 95\% confidence interval for the true mean reduction in drying time.

1 In the past, the times for workers in a factory to complete a particular task had a known median of 7.4 minutes. Following a review, managers at the factory wish to know if the median time to complete the task has been reduced.

1. A random sample of 12 times, in minutes, gives the following results. $$\begin{array} { l l l l l l l l l l l l } 6.90 & 7.23 & 6.54 & 7.62 & 7.04 & 7.33 & 6.74 & 6.45 & 7.81 & 7.71 & 7.50 & 6.32 \end{array}$$ Carry out an appropriate test using a \(5 \%\) level of significance.
2. Some time later, a much larger random sample of times gives the following results. $$n = 80 \quad \sum x = 555.20 \quad \sum x ^ { 2 } = 3863.9031$$ Find a \(95 \%\) confidence interval for the true mean time for the task. Justify your choice of which distribution to use.
3. Describe briefly one advantage and one disadvantage of having a \(99 \%\) confidence interval instead of a \(95 \%\) confidence interval.

2 A company supplying cattle feed to dairy farmers claims that its new brand of feed will increase average milk yields by 10 litres per cow per week. A farmer thinks the increase will be less than this and decides to carry out a statistical investigation using a paired \(t\) test. A random sample of 10 dairy cows are given the new feed and then their milk yields are compared with their yields when on the old feed. The yields, in litres per week, for the 10 cows are as follows.

1. Why is it sensible to use a paired test?
2. State the condition necessary for a paired \(t\) test.
3. Assuming the condition stated in part (ii) is met, carry out the test, using a significance level of \(5 \%\), to see whether it appears that the company's claim is justified.
4. Find a 95\% confidence interval for the mean increase in the milk yield using the new feed.

2

1. Explain what is meant by a simple random sample. A manufacturer produces tins of paint which nominally contain 1 litre. The quantity of paint delivered by the machine that fills the tins can be assumed to be a Normally distributed random variable. The machine is designed to deliver an average of 1.05 litres to each tin. However, over time paint builds up in the delivery nozzle of the machine, reducing the quantity of paint delivered. Random samples of 10 tins are taken regularly from the production process. If a significance test, carried out at the \(5 \%\) level, suggests that the average quantity of paint delivered is less than 1.02 litres, the machine is cleaned.
2. By carrying out an appropriate test, determine whether or not the sample below leads to the machine being cleaned. $$\begin{array} { l l l l l l l l l l } 0.994 & 1.010 & 1.021 & 1.015 & 1.016 & 1.022 & 1.009 & 1.007 & 1.011 & 1.026 \end{array}$$ Each time the machine has been cleaned, a random sample of 10 tins is taken to determine whether or not the average quantity of paint delivered has returned to 1.05 litres.
3. On one occasion after the machine has been cleaned, the quality control manager thinks that the distribution of the quantity of paint is symmetrical but not necessarily Normal. The sample on this occasion is as follows.

   1.055	1.064	1.063	1.043	1.062	1.070	1.059	1.044	1.054
   1.053

   By carrying out an appropriate test at the \(5 \%\) level of significance, determine whether or not this sample supports the conclusion that the average quantity of paint delivered is 1.05 litres.

3

1. A personal trainer believes that drinking a glass of beetroot juice an hour before exercising enables endurance tests to be completed more quickly. To test his belief he takes a random sample of 12 of his trainees and, on two occasions, asks them to carry out 100 repetitions of a particular exercise as quickly as possible. Each trainee drinks a glass of water on one occasion and a glass of beetroot juice on the other occasion. The times in seconds taken by the trainees are given in the table.

   Trainee	Water	Beetroot juice
   A	75.1	72.9
   B	86.2	79.9
   C	77.3	71.6
   D	89.1	90.2
   E	67.9	68.2
   F	101.5	95.2
   G	82.5	76.5
   H	83.3	80.2
   I	102.5	99.1
   J	91.3	82.2
   K	92.5	90.1
   L	77.2	77.9

   The trainer wishes to test his belief using a paired \(t\) test at the \(1 \%\) level of significance. Assuming any necessary assumptions are valid, carry out a test of the hypotheses \(\mathrm { H } _ { 0 } : \mu _ { D } = 0 , \mathrm { H } _ { 1 } : \mu _ { D } < 0\), where \(\mu _ { D }\) is the population mean difference in times (time with beetroot juice minus time with water).
2. An ornithologist believes that the number of birds landing on the bird feeding station in her garden in a given interval of time during the morning should follow a Poisson distribution. In order to test her belief, she makes the following observations in 60 randomly chosen minutes one morning.

   Number of birds	0	1	2	3	4	5	6	\(\geqslant 7\)
   Frequency	2	5	10	17	14	7	4	1

   Given that the data in the table have a mean value of 3.3, use a goodness of fit test, with a significance level of \(5 \%\), to investigate whether the ornithologist is justified in her belief.

4 An insurance company is investigating a new system designed to reduce the average time taken to process claim forms. The company has decided to use 10 experienced employees to process claims using the old system and the new system. Two procedures for comparing the systems are proposed.
Procedure \(A\) There are two sets of claim forms, set 1 and set 2. Each contains the same number of forms. Each employee processes set 1 on the old system and set 2 on the new system. The times taken are compared. Procedure \(B\) There is just one set of claim forms which each employee processes firstly on the old system and then on the new system. The times taken are compared.

1. State one weakness of each of these procedures. In fact a third procedure which avoids these two weaknesses is adopted. In this procedure each employee is given a randomly selected set of claim forms. Each set contains the same number of forms. The employees each process their set of claim forms on both systems. The times taken, in minutes, are shown in the table.

   Employee	1	2	3	4	5	6	7	8	9	10
   Old system	40.5	42.9	52.8	51.7	77.2	66.7	65.2	49.2	55.6	58.3
   New system	39.2	40.7	50.6	50.7	71.4	70.5	71.1	47.7	52.1	55.5

2. Carry out a paired \(t\) test at the \(5 \%\) level of significance to investigate whether the mean length of time taken to process a set of forms has reduced using the new system.
3. State fully the usual conditions for a paired \(t\) test.
4. Construct a \(99 \%\) confidence interval for the mean reduction in time taken to process a set of forms using the new system.

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{cases}$$ where \(\theta\) is an unknown positive constant.

1. Find \(\mathrm { E } \left( X ^ { n } \right)\), where \(n \neq - 2\), and hence write down the value of \(\mathrm { E } ( X )\).
2. Find
   1. \(\operatorname { Var } ( X )\),
   2. \(\operatorname { Var } \left( X ^ { 2 } \right)\).
   3. Find \(\mathrm { E } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)\) and \(\mathrm { E } \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } + X _ { 3 } ^ { 2 } \right)\), where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\). Hence construct unbiased estimators, \(T _ { 1 }\) and \(T _ { 2 }\), of \(\theta\) and \(\operatorname { Var } ( X )\) respectively, which are based on \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
   4. Find \(\operatorname { Var } \left( T _ { 2 } \right)\).