Single sample t-test

There are a large number of students at a particular college. The heights, in metres, of a random sample of 8 students are as follows. $$1.75 \quad 1.72 \quad 1.62 \quad 1.70 \quad 1.82 \quad 1.75 \quad 1.68 \quad 1.84$$ You may assume that heights of students are normally distributed.

1. Test, at the 5\% significance level, whether the population mean height of students at this college is greater than 1.70 metres. [7]
2. Find a 95\% confidence interval for the population mean height of students at this college. [3]

The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 20\). He weighs a random sample of 6 bags and the results are summarised as follows: $$\Sigma x = 430 \quad \Sigma x^2 = 40$$ Carry out the test at the 5\% significance level. [7]

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, \(x\) kg, of apples produced in a year are summarised as follows. $$\sum x = 152.0 \qquad \sum x^2 = 2313.0$$ Test, at the 5% significance level, whether Farmer B's claim is justified. [6]

Judith, the village postmistress, believes that, since moving the post office counter into the local pharmacy, the mean daily number of customers that she serves has increased from \(79\). In order to investigate her belief, she counts the number of customers that she serves on \(12\) randomly selected days, with the following results. \(88 \quad 81 \quad 84 \quad 89 \quad 90 \quad 77 \quad 72 \quad 80 \quad 82 \quad 81 \quad 75 \quad 85\) Stating a necessary distributional assumption, test Judith's belief at the \(5\%\) level of significance. [9 marks]

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]

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 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]

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)

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]

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]

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]

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]

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 retailer sells bags of flour which are advertised as containing 1.5 kg of flour. A trading standards officer is investigating whether there is enough flour in each bag. He collects a random sample and uses software to carry out a hypothesis test at the 5\% level. The analysis is shown in the software printout below. \includegraphics{figure_12}

1. State the hypotheses the officer uses in the test, defining any parameters used. [2]
2. State the distribution used in the analysis. [3]
3. Carry out the hypothesis test, giving your conclusion in context. [3]

A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams. The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average. A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are $$\sum x = 246 \quad \text{and} \quad \sum x^2 = 10198$$ Investigate the shopkeeper's claim using the 5\% level of significance. State any assumptions that you make. [9 marks]