T-tests (unknown variance)

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.

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]

1. At the start of each academic year, a large college carries out a diagnostic test on a random sample of new students. Past experience has shown that the standard deviation of the scores on this test is 19.71

The admissions tutor claimed that the new students in 2013 would have more varied scores than usual. The scores for the students taking the test can be assumed to come from a normal distribution. A random sample of 10 new students was taken and the score \(x\), for each student was recorded. The data are summarised as \(\sum x = 619 \sum x ^ { 2 } = 42397\)

1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test the admission tutor's claim. The admissions tutor decides that in future he will use the same hypotheses but take a larger sample of size 30 and use a significance level of 1\%.
2. Use the tables to show that, to 3 decimal places, the critical region for \(S ^ { 2 }\) is \(S ^ { 2 } > 664.281\)
3. Find the probability of a type II error using this test when the true value of the standard deviation is in fact 22.20

1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4

1 There are 18 people in Millie's class. To choose a person at random she numbers the people in the class from 1 to 18 and presses the random number button on her calculator to obtain a 3-digit decimal. Millie then multiplies the first digit in this decimal by two and chooses the person corresponding to this new number. Decimals in which the first digit is zero are ignored.

1. Give a reason why this is not a satisfactory method of choosing a person. Millie obtained a random sample of 5 people of her own age by a satisfactory sampling method and found that their heights in metres were \(1.66,1.68,1.54,1.65\) and 1.57 . Heights are known to be normally distributed with variance \(0.0052 \mathrm {~m} ^ { 2 }\).
2. Find a \(98 \%\) confidence interval for the mean height of people of Millie's age.

7. A doctor wishes to study the level of blood glucose in males. The level of blood glucose is normally distributed. The doctor measured the blood glucose of 10 randomly selected male students from a school. The results, in mmol/litre, are given below. $$\begin{array} { l l l l l l l l l l } 4.7 & 3.6 & 3.8 & 4.7 & 4.1 & 2.2 & 3.6 & 4.0 & 4.4 & 5.0 \end{array}$$

1. Calculate a \(95 \%\) confidence interval for the mean.
2. Calculate a 95\% confidence interval for the variance. A blood glucose reading of more than 7 mmol/litre is counted as high.
3. Use appropriate confidence limits from parts (a) and (b) to find the highest estimate of the proportion of male students in the school with a high blood glucose level. \section*{END}

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]

1 A manager is investigating the times taken by employees to complete a particular task as a result of the introduction of new technology. He claims that the mean time taken to complete the task is reduced by more than 0.4 minutes. He chooses a random sample of 10 employees. The times taken, in minutes, before and after the introduction of the new technology are recorded in the table.

1. Test at the 10\% significance level whether the manager's claim is justified.
2. State an assumption that is necessary for this test to be valid.

4 Manet has developed a new training course to help athletes improve their time taken to run 800 m . Manet claims that his course will decrease an athlete's time by more than 2 s on average. For a random sample of 10 athletes the times taken, in seconds, before and after the course are given in the following table. Use a \(t\)-test, at the \(5 \%\) significance level, to test whether Manet's claim is justified, stating any assumption that you make.

Two fish farmers \(X\) and \(Y\) produce a particular type of fish. Farmer \(X\) chooses a random sample of 8 of his fish and records the masses, \(x\) kg, as follows. 1.2 \quad 1.4 \quad 0.8 \quad 2.1 \quad 1.8 \quad 2.6 \quad 1.5 \quad 2.0 Farmer \(Y\) chooses a random sample of 10 of his fish and summarises the masses, \(y\) kg, as follows. $$\Sigma y = 20.2 \quad \Sigma y^2 = 44.6$$ You should assume that both distributions are normal with equal variances. Test at the 10% significance level whether the mean mass of fish produced by farmer \(X\) differs from the mean mass of fish produced by farmer \(Y\). [10]

Answer only one of the following two alternatives. **EITHER** One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N, is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses 2 kg and \(M\) kg attached to its free end. The \(M\) kg particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards. \begin{enumerate}[label=(\roman*)] \item Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period \(\frac{2\pi}{5}\) s. State the distance below \(O\) of the centre of the oscillations. [7] \item The speed of the 2 kg particle is 0.4 m s\(^{-1}\) when its displacement from the centre of oscillation is 0.06 m. Find the amplitude of the motion. [3] \item Deduce the value of \(M\). [4] \end{enumerate] **OR** In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows. \(\Sigma x = 10.56\) \(\quad\) \(\Sigma x^2 = 14.1775\) \(\quad\) \(\Sigma y = 12.39\) \(\quad\) \(\Sigma y^2 = 15.894\) A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\). \begin{enumerate}[label=(\roman*)] \item Test, at the 10% significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal. [9] \item A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as \(\Sigma x = 10.24\) \(\quad\) and \(\quad\) \(\Sigma(x - \bar{x})^2 = 0.294\), where \(\bar{x}\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p\) kg. A test of this claim is carried out at the 10% significance level, using only this second sample from lake \(A\). This test supports the scientist's claim. Find the greatest possible value of \(p\). [5] \end{enumerate]

9 A gardener \(P\) claims that a new type of fruit tree produces a higher annual mass of fruit than the type that he has previously grown. The old type of tree produced 5.2 kg of fruit per tree, on average. A random sample of 10 trees of the new type is chosen. The masses, \(x \mathrm {~kg}\), of fruit produced are summarised as follows. $$\Sigma x = 61.0 \quad \Sigma x ^ { 2 } = 384.0$$ Test, at the \(5 \%\) significance level, whether gardener \(P\) 's claim is justified, assuming a normal distribution. Another gardener \(Q\) has his own type of fruit tree. The masses, \(y \mathrm {~kg}\), of fruit produced by a random sample of 10 trees grown by gardener \(Q\) are summarised as follows. $$\Sigma y = 70.0 \quad \Sigma y ^ { 2 } = 500.6$$ Test, at the \(5 \%\) significance level, whether the mean mass of fruit produced by gardener \(Q\) 's trees is greater than the mean mass of fruit produced by gardener \(P\) 's trees. You may assume that both distributions are normal and you should state any additional assumption.

9 Experiments are conducted to test the breaking strength of each of two types of rope, \(P\) and \(Q\). A random sample of 50 ropes of type \(P\) and a random sample of 70 ropes of type \(Q\) are selected. The breaking strengths, \(p\) and \(q\), measured in appropriate units, are summarised as follows. $$\Sigma p = 321.2 \quad \Sigma p ^ { 2 } = 2120.0 \quad \Sigma q = 475.3 \quad \Sigma q ^ { 2 } = 3310.0$$ Test, at the \(10 \%\) significance level, whether the mean breaking strengths of type \(P\) and type \(Q\) ropes are the same.

2 As part of a comparison of two varieties of cucumber, Fanfare and Marketmore, random samples of harvested cucumbers of each variety were selected and their lengths measured, in centimetres. The results are summarised in the table.

1. Test, at the \(1 \%\) level of significance, the hypothesis that there is no difference between the mean length of harvested Fanfare cucumbers and that of harvested Marketmore cucumbers.
2. In addition to length, name one other characteristic of cucumbers that could be used for comparative purposes.

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 \(n\).

6 The random variable \(X\) was assumed to have a normal distribution with mean \(\mu\). Using a random sample of size 128, a significance test was carried out using the following hypotheses. \(\mathrm { H } _ { 0 } : \mu = 30\) \(\mathrm { H } _ { 1 } : \mu > 30\) It was found that \(\sum x = 3929.6\) and \(\sum x ^ { 2 } = 123483.52\). The conclusion of the test was to reject the null hypothesis.

1. Determine the range of possible values of the significance level of the test.
2. It was subsequently found that \(X\) was not normally distributed. Explain whether this invalidates the conclusion of the test.

6 A company has two machines, \(A\) and \(B\), which independently fill small bottles with a liquid. The volumes of liquid per bottle, in suitable units, filled by machines \(A\) and \(B\) are denoted by \(x\) and \(y\) respectively. A scientist at the company takes a random sample of 40 bottles filled by machine \(A\) and a random sample of 50 bottles filled by machine \(B\). The results are summarised as follows. $$\sum x = 1120 \quad \sum x ^ { 2 } = 31400 \quad \sum y = 1370 \quad \sum y ^ { 2 } = 37600$$ The population means of the volumes of liquid in the bottles filled by machines \(A\) and \(B\) are denoted by \(\mu _ { A }\) and \(\mu _ { B }\).

1. Test at the \(2 \%\) significance level whether there is any difference between \(\mu _ { A }\) and \(\mu _ { B }\).
2. Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { \mathrm { A } } - \mu _ { \mathrm { B } }\) is greater than 0.25.
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The times taken, in hours, by cyclists from two different clubs, \(A\) and \(B\), to complete a 50 km time trial are being compared. The times taken by a cyclist from club \(A\) and by a cyclist from club \(B\) are denoted by \(t _ { A }\) and \(t _ { B }\) respectively. A random sample of 50 cyclists from \(A\) and a random sample of 60 cyclists from \(B\) give the following summarised data. $$\Sigma t _ { A } = 102.0 \quad \Sigma t _ { A } ^ { 2 } = 215.18 \quad \Sigma t _ { B } = 129.0 \quad \Sigma t _ { B } ^ { 2 } = 282.3$$ Using a 5\% significance level, test whether, on average, cyclists from club \(A\) take less time to complete the time trial than cyclists from club \(B\). A test at the \(\alpha \%\) significance level shows that there is evidence that the population mean time for cyclists from club \(B\) exceeds the population mean time for cyclists from club \(A\) by more than 0.05 hours. Find the set of possible values of \(\alpha\).

Sweet pea plants grown using a standard plant food have a mean height of 1.6 m. A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$n = 49$$ $$\sum x = 74.48$$ $$\sum x^2 = 120.8896$$ Test, at the 5\% significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m. [9]

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.

The independent variables \(X\) and \(Y\) have distributions with the same variance \(\sigma^2\). Random samples of \(N\) observations of \(X\) and \(2N\) observations of \(Y\) are taken, and the results are summarised by $$\Sigma x = 4, \quad \Sigma x^2 = 10, \quad \Sigma y = 8, \quad \Sigma y^2 = 102.$$ These data give a pooled estimate of \(10\) for \(\sigma^2\). Find \(N\). [5]

A particle of mass \(m\) is attached to one end \(A\) of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and the particle hangs in equilibrium under gravity. The particle is projected horizontally so that it starts to move in a vertical circle. The string slackens after turning through an angle of \(120°\). Show that the speed of the particle is then \(\sqrt{\left(\frac{4}{3}ga\right)}\) and find the initial speed of projection. [5]

8 A company decides that its employees should follow an exercise programme for 30 minutes each day, with the aim that they lose weight and increase productivity. The weights, in kg , of a random sample of 8 employees at the start of the programme and after following the programme for 6 weeks are shown in the table. Assuming that loss in weight is normally distributed, find a 95\% confidence interval for the mean loss in weight of the company's employees. Test at the \(5 \%\) significance level whether, after the exercise programme, there is a reduction of more than 2.5 kg in the population mean weight.

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.

3 A machine produces circular metal discs whose radii have a normal distribution with mean \(\mu \mathrm { cm }\). A random sample of five discs is selected and their radii, in cm, are as follows. $$\begin{array} { l l l l l } 6.47 & 6.52 & 6.46 & 6.47 & 6.51 \end{array}$$

1. Calculate a \(95 \%\) confidence interval for \(\mu\).
2. Hence state a 95\% confidence interval for the mean circumference of a disc.