One-sample t-test, variance unknown

4 The mean time to mark a certain set of examination papers is estimated by the examination board to be 12 minutes per paper. A random sample of 150 examination papers gave \(\Sigma x = 2130\) and \(\Sigma x ^ { 2 } = 37746\), where \(x\) is the time in minutes to mark an examination paper.

1. Calculate unbiased estimates of the population mean and variance.
2. Stating the null and alternative hypotheses, use a \(10 \%\) significance level to test whether the examination board's estimated time is consistent with the data.

4

1. A random sample of 8 boxes of cereal from a certain supplier was taken. Each box was weighed and the masses in grams were as follows. $$\begin{array} { l l l l l l l l } 261 & 249 & 259 & 252 & 255 & 256 & 258 & 254 \end{array}$$ Find unbiased estimates of the population mean and variance.
2. The supplier claims that the mean mass of boxes of cereal is 253 g . A quality control officer suspects that the mean mass is actually more than 253 g . In order to test this claim, he weighs a random sample of 100 boxes of cereal and finds that the total mass is 25360 g .
   1. Given that the population standard deviation of the masses is 3.5 g , test at the \(5 \%\) significance level whether the population mean mass is more than 253 g .
      An employee says, 'This test is invalid because it uses the normal distribution, but we do not know whether the masses of the boxes are normally distributed.’
   2. Explain briefly whether this statement is true or not.

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.

5 A large number of children are competing in a throwing competition. The distances, in metres, thrown by a random sample of 8 children are as follows. \(\begin{array} { l l l l l l l l } 19.8 & 22.1 & 24.4 & 21.5 & 20.8 & 26.3 & 23.7 & 25.0 \end{array}\)

1. Assuming that distances are normally distributed, test, at the \(5 \%\) significance level, whether the population mean distance thrown is more than 22.0 metres.
2. Find a 95\% confidence interval for the population mean distance thrown.

1 Farmer \(A\) grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer \(B\) grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer \(A\) 's trees. The masses of apples from Farmer \(B\) 's trees may be assumed to be normally distributed. A random sample of 10 trees from Farmer \(B\) is chosen. The masses, \(x \mathrm {~kg}\), of apples produced in a year are summarised as follows. $$\sum x = 152.0 \quad \sum x ^ { 2 } = 2313.0$$ Test, at the \(5 \%\) significance level, whether Farmer \(B\) 's claim is justified.

5 Raman is researching the heights of male giraffes in a particular region. Raman assumes that the heights of male giraffes in this region are normally distributed. He takes a random sample of 8 male giraffes from the region and measures the height, in metres, of each giraffe. These heights are as follows. $$\begin{array} { c c c c c c c c } 5.2 & 5.8 & 4.9 & 6.1 & 5.5 & 5.9 & 5.4 & 5.6 \end{array}$$

1. Find a \(90 \%\) confidence interval for the population mean height of male giraffes in this region. [5]
   Raman claims that the population mean height of male giraffes in the region is less than 5.9 metres.
2. Test at the \(2.5 \%\) significance level whether this sample provides sufficient evidence to support Raman's claim.

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.

4 The height of sweet pea plants grown in a nursery is a random variable. A random sample of 50 plants is measured and is found to have a mean height 1.72 m and variance \(0.0967 \mathrm {~m} ^ { 2 }\).

1. Calculate an unbiased estimate for the population variance of the heights of sweet pea plants.
2. Hence test, at the \(10 \%\) significance level, whether the mean height of sweet pea plants grown by the nursery is 1.8 m , stating your hypotheses clearly.

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.

4 A machine is set to produce metal discs with mean diameter 15.4 mm . In order to test the correctness of the setting, a random sample of 12 discs was selected and the diameters, \(x \mathrm {~mm}\), were measured. The results are summarised by \(\Sigma x = 177.6\) and \(\Sigma x ^ { 2 } = 2640.40\). Diameters may be assumed to be normally distributed with mean \(\mu \mathrm { mm }\).

1. Find a \(95 \%\) confidence interval for \(\mu\).
2. Test, at the \(5 \%\) significance level, the null hypothesis \(\mu = 15.4\) against the alternative hypothesis \(\mu < 15.4\).

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.

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.

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.

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.

9 A random sample of 8 observations of a normal random variable \(X\) gave the following summarised data, where \(\bar { x }\) denotes the sample mean. $$\Sigma x = 42.5 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 15.519$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is greater than 4.5. Calculate a 95\% confidence interval for the population mean of \(X\).

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.

8 A large number of long jumpers are competing in a national long jump competition. The distances, in metres, jumped by a random sample of 7 competitors are as follows. $$\begin{array} { l l l l l l l } 6.25 & 7.01 & 5.74 & 6.89 & 7.24 & 5.64 & 6.52 \end{array}$$ Assuming that distances are normally distributed, test, at the \(5 \%\) significance level, whether the mean distance jumped by long jumpers in this competition is greater than 6.2 metres. The distances jumped by another random sample of 8 long jumpers in this competition are recorded. Using the data from this sample of 8 long jumpers, a \(95 \%\) confidence interval for the population mean, \(\mu\) metres, is calculated as \(5.89 < \mu < 6.75\). Find the unbiased estimates for the population mean and population variance used in this calculation.

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.