Single sample t-test

1 A random sample of 7 observations of a variable \(X\) are as follows. $$\begin{array} { l l l l l l l } 8.26 & 7.78 & 7.92 & 8.04 & 8.27 & 7.95 & 8.34 \end{array}$$ The population mean of \(X\) is \(\mu\).

1. Test, at the \(10 \%\) significance level, the null hypothesis \(\mu = 8.22\) against the alternative hypothesis \(\mu < 8.22\).
2. State an assumption necessary for the test in part (a) to be valid.

1 The heights of the members of a large sports club are normally distributed. A random sample of 11 members of the club is chosen and their heights, \(x \mathrm {~cm}\), are measured. The results are summarised as follows, where \(\bar { x }\) denotes the sample mean of \(x\). $$\bar { x } = 176.2 \quad \sum ( x - \bar { x } ) ^ { 2 } = 313.1$$ Test, at the \(5 \%\) significance level, the null hypothesis that the population mean height for members of this club is equal to 172.5 cm against the alternative hypothesis that the mean differs from 172.5 cm . [5]

1 Maya is an athlete who competes in 1500-metre races. Last summer her practice run times had mean 4.22 minutes. Over the winter she has done some intense training to try to improve her times. A random sample of 10 of her practice run times, \(x\) minutes, this summer are summarised as follows. $$\sum x = 42.05 \quad \sum x ^ { 2 } = 176.83$$ Maya's new practice run times are normally distributed. She believes that on average her times have improved as a result of her training. Test, at the \(5 \%\) significance level, whether Maya's belief is supported by the data.

2 A particular type of engine used in rockets is designed to have a mean lifetime of at least 2000 hours. A check of four randomly chosen engines yielded the following lifetimes in hours. $$\begin{array} { l l l l } 1896.4 & 2131.5 & 1903.3 & 1901.6 \end{array}$$ A significance test of whether engines meet the design is carried out. It may be assumed that lifetimes have a normal distribution.

1. Give a reason why a \(t\)-test should be used.
2. Carry out the test at the \(10 \%\) significance level.

2 The manager of a large country estate is preparing to plant an area of woodland. He orders a large number of saplings (young trees) from a nursery. He selects a random sample of 12 of the saplings and measures their heights, which are as follows (in metres). $$\begin{array} { l l l l l l l l l l l l } 0.63 & 0.62 & 0.58 & 0.56 & 0.59 & 0.62 & 0.64 & 0.58 & 0.55 & 0.61 & 0.56 & 0.52 \end{array}$$

1. The manager requires that the mean height of saplings at planting is at least 0.6 metres. Carry out the usual \(t\) test to examine this, using a \(5 \%\) significance level. State your hypotheses and conclusion carefully. What assumption is needed for the test to be valid?
2. Find a \(95 \%\) confidence interval for the true mean height of saplings. Explain carefully what is meant by a \(95 \%\) confidence interval.
3. Suppose the assumption needed in part (i) cannot be justified. Identify an alternative test that the manager could carry out in order to check that the saplings meet his requirements, and state the null hypothesis for this test.

3 An athlete finds that her times for running 100 m are normally distributed. Before a period of intensive training, her mean time is 11.8 s . After the period of intensive training, five randomly selected times, in seconds, are as follows. $$\begin{array} { l l l l l } 11.70 & 11.65 & 11.80 & 11.75 & 11.60 \end{array}$$ Carry out a suitable test, at the \(5 \%\) significance level, to investigate whether times after the training are less, on average, than times before the training.

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.

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.

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?

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.

7 A greengrocer claims that his cabbages have a mean mass of more than 1.2 kg . In order to check his claim, he weighs 10 cabbages, chosen at random from his stock. The masses, in kg, are as follows. $$\begin{array} { l l l l l l l l l l } 1.26 & 1.24 & 1.17 & 1.23 & 1.18 & 1.25 & 1.19 & 1.20 & 1.21 & 1.18 \end{array}$$ Stating any assumption that you make, test at the \(10 \%\) significance level whether the greengrocer's claim is supported by this evidence.

6 A random sample of 10 observations of a normal random variable \(X\) has mean \(\bar { x }\), where $$\bar { x } = 8.254 , \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.912 .$$ Using a \(5 \%\) significance level, test whether the mean of \(X\) is greater than 8.05.

7 A random sample of 9 observations of a normal variable \(X\) is taken. The results are summarised as follows. $$\Sigma x = 24.6 \quad \Sigma x ^ { 2 } = 68.5$$ Test, at the \(5 \%\) significance level, whether the population mean is greater than 2.5.

7 A large number of athletes are taking part in a competition. The masses, in kg , of a random sample of 7 athletes are as follows. $$\begin{array} { l l l l l l l } 98.1 & 105.0 & 92.2 & 89.8 & 99.9 & 95.4 & 101.2 \end{array}$$ Assuming that masses are normally distributed, test, at the \(10 \%\) significance level, whether the mean mass of athletes in this competition is equal to 94 kg .

7 A random sample of 10 observations of a normally distributed random variable \(X\) gave the following summarised data, where \(\bar { x }\) denotes the sample mean. $$\Sigma x = 70.4 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 8.48$$ Test, at the \(10 \%\) significance level, whether the population mean of \(X\) is less than 7.5.

6 A random sample of 8 observations of a normal random variable \(X\) has mean \(\bar { x }\), where $$\bar { x } = 6.246 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.784$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is less than 6.44.

3 Engineers in charge of a chemical plant need to monitor the temperature inside a reaction chamber. Past experience has shown that when functioning correctly the temperature inside the chamber can be modelled by a Normal distribution with mean \(380 ^ { \circ } \mathrm { C }\). The engineers are concerned that the mean operating temperature may have fallen. They decide to test the mean using the following random sample of 12 recent temperature readings.

1. Give three reasons why a \(t\) test would be appropriate.
2. Carry out the test using a \(5 \%\) significance level. State your hypotheses and conclusion carefully.
3. Find a 95\% confidence interval for the true mean temperature in the reaction chamber.
4. Describe briefly one advantage and one disadvantage of having a 99\% confidence interval instead of a 95\% confidence interval.

8 Bottles of sherry nominally contain 1000 millilitres. After the introduction of a new method of filling the bottles, there is a suspicion that the mean volume of sherry in a bottle has changed. In order to investigate this suspicion, a random sample of 12 bottles of sherry is taken and the volume of sherry in each bottle is measured. The volumes, in millilitres, of sherry in these bottles are found to be Assuming that the volume of sherry in a bottle is normally distributed, investigate, at the \(5 \%\) level of significance, whether the mean volume of sherry in a bottle differs from 1000 millilitres.

5 Jasmine's French teacher states that a homework assignment should take, on average, 30 minutes to complete. Jasmine believes that he is understating the mean time that the assignment takes to complete and so decides to investigate. She records the times, in minutes, that it takes for a random sample of 10 students to complete the French assignment, with the following results: $$\begin{array} { l l l l l l l l l l } 29 & 33 & 36 & 42 & 30 & 28 & 31 & 34 & 37 & 35 \end{array}$$

1. Test, at the \(1 \%\) level of significance, Jasmine's belief that her French teacher has understated the mean time that it should take to complete the homework assignment.
2. State an assumption that you must make in order for the test used in part (a) to be valid.

3 Lorraine bought a new golf club. She then practised with this club by using it to hit golf balls on a golf range. After several such practice sessions, she believed that there had been no change from 190 metres in the mean distance that she had achieved when using her old club. To investigate this belief, she measured, at her next practice session, the distance, \(x\) metres, of each of a random sample of 10 shots with her new club. Her results gave $$\sum x = 1840 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1240$$ Investigate Lorraine's belief at the \(2 \%\) level of significance, stating any assumption that you make.
(7 marks)

6 The contents, in millilitres, of cartons of milk produced at Kream Dairies, can be modelled by a normal distribution with mean 568 and variance \(\sigma ^ { 2 }\). After receiving several complaints from their customers who thought that the average content of the cartons had been reduced, the production manager of Kream Dairies decided to investigate. A random sample of 8 cartons of milk was taken, revealing the following contents, in millilitres. $$\begin{array} { l l l l l l l l } 560 & 568 & 561 & 562 & 564 & 567 & 565 & 563 \end{array}$$ Investigate, at the \(1 \%\) level of significance, whether the average content of cartons of milk is less than 568 millilitres.
(10 marks)

6 The lifetime, \(X\) hours, of Everwhite camera batteries is normally distributed. The manufacturer claims that the mean lifetime of these batteries is 100 hours.

1. The members of a photography club suspect that the batteries do not last as long as is claimed by the manufacturer. In order to investigate their suspicion, the members test a random sample of five of these batteries and find the lifetimes, in hours, to be as follows: $$\begin{array} { l l l l l } 85 & 92 & 100 & 95 & 99 \end{array}$$ Test the members' suspicion at the \(5 \%\) level of significance.
2. The manufacturer, believing that the mean lifetime of these batteries has not changed from 100 hours, decides to determine the lifetime, \(x\) hours, of each of a random sample of 80 Everwhite camera batteries. The manufacturer obtains the following results, where \(\bar { x }\) denotes the sample mean: $$\sum x = 8080 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 6399$$ Test the manufacturer's belief at the \(5 \%\) level of significance.

6 The management of the Wellfit gym claims that the mean cholesterol level of those members who have held membership of the gym for more than one year is 3.8 . A local doctor believes that the management's claim is too low and investigates by measuring the cholesterol levels of a random sample of 7 such members of the Wellfit gym, with the following results: $$\begin{array} { l l l l l l l } 4.2 & 4.3 & 3.9 & 3.8 & 3.6 & 4.8 & 4.1 \end{array}$$ Is there evidence, at the \(5 \%\) level of significance, to justify the doctor's belief that the mean cholesterol level is greater than the management's claim? State any assumption that you make.

6 South Riding Alarms (SRA) maintains household burglar-alarm systems. The company aims to carry out an annual service of a system in a mean time of 20 minutes.
Technicians who carry out an annual service must record the times at which they start and finish the service.

1. Gary is employed as a technician by SRA and his manager, Rajul, calculates the times taken for 8 annual services carried out by Gary. The results, in minutes, are as follows. $$\begin{array} { l l l l l l l l } 24 & 25 & 29 & 16 & 18 & 27 & 19 & 23 \end{array}$$ Assume that these times may be regarded as a random sample from a normal distribution. Carry out a hypothesis test, at the \(10 \%\) significance level, to examine whether the mean time for an annual service carried out by Gary is 20 minutes.
   [0pt] [8 marks]
2. Rajul suspects that Gary may be taking longer than 20 minutes on average to carry out an annual service. Rajul therefore calculates the times taken for 100 annual services carried out by Gary. Assume that these times may also be regarded as a random sample from a normal distribution but with a standard deviation of 4.6 minutes. Find the highest value of the sample mean which would not support Rajul's suspicion at the \(5 \%\) significance level. Give your answer to two decimal places.
   [0pt] [4 marks] \(7 \quad\) A continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 4 } { 5 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 20 } ( x - 3 ) ( 3 x - 11 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
   1. Find \(\mathrm { P } ( X < 1 )\).
   2. 1. Show that, for \(1 \leqslant x \leqslant 3\), the cumulative distribution function, \(\mathrm { F } ( x )\), is given by $$\mathrm { F } ( x ) = \frac { 1 } { 20 } \left( x ^ { 3 } - 10 x ^ { 2 } + 33 x - 16 \right)$$
      2. Hence verify that the median value of \(X\) lies between 1.13 and 1.14 .
         [0pt] [3 marks] QUESTION
         PART Answer space for question 7
         REFERENCE REFERENCE
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2. 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 minutes \({ } ^ { 2 }\). 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)