Two-sample t-test with summary statistics

5 Students at two colleges, \(A\) and \(B\), are competing in a computer games challenge.

1. The time taken for a randomly chosen student from college \(A\) to complete the challenge has a normal distribution with mean \(\mu\) minutes. The times taken, \(x\) minutes, are recorded for a random sample of 10 students chosen from college \(A\). The results are summarised as follows. $$\sum x = 828 \quad \sum x ^ { 2 } = 68622$$ A test is carried out on the data at the \(5 \%\) significance level and the result supports the claim that \(\mu > k\). Find the greatest possible value of \(k\).
2. A random sample of 8 students is chosen from college \(B\). Their times to complete the same challenge give a sample mean of 79.8 minutes and an unbiased variance estimate of 9.966 minutes \({ } ^ { 2 }\). Use a 2 -sample test at the \(5 \%\) significance level to test whether the mean time for students at college \(B\) to complete the challenge is the same as the mean time for students at college \(A\) to complete the challenge. You should assume that the two distributions are normal and have the same population variance.

4 A scientist is investigating the lengths of the leaves of birch trees in different regions. He takes a random sample of 50 leaves from birch trees in region \(A\) and a random sample of 60 leaves from birch trees in region \(B\). He records their lengths in \(\mathrm { cm } , x\) and \(y\), respectively. His results are summarised as follows. $$\sum x = 282 \quad \sum x ^ { 2 } = 1596 \quad \sum y = 328 \quad \sum y ^ { 2 } = 1808$$ The population mean lengths of leaves from birch trees in regions \(A\) and \(B\) are \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively. Carry out a test at the \(5 \%\) significance level to test the null hypothesis \(\mu _ { \mathrm { A } } = \mu _ { \mathrm { B } }\) against the alternative hypothesis \(\mu _ { \mathrm { A } } \neq \mu _ { \mathrm { B } }\).

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.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

4 An inspector is checking the lengths of metal rods produced by two machines, \(X\) and \(Y\). These rods should be of the same length, but the inspector suspects that those made by machine \(X\) are shorter, on average, than those made by machine \(Y\). The inspector chooses a random sample of 80 rods made by machine \(X\) and a random sample of 60 rods made by machine \(Y\). The lengths of these rods are \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively. Her results are summarised as follows. $$\sum x = 164.0 \quad \sum x ^ { 2 } = 338.1 \quad \sum y = 124.8 \quad \sum y ^ { 2 } = 261.1$$

1. Test at the \(10 \%\) significance level whether the data supports the inspector's suspicion.
2. Give a reason why it is not necessary to make any assumption about the distributions of the lengths of the rods.

3 A scientist is investigating the masses of birds of a certain species in country \(X\) and country \(Y\). She takes a random sample of 50 birds of this species from country \(X\) and a random sample of 80 birds of this species from country \(Y\). She records their masses in \(\mathrm { kg } , x\) and \(y\), respectively. Her results are summarised as follows. $$\sum x = 75.5 \quad \sum x ^ { 2 } = 115.2 \quad \sum y = 116.8 \quad \sum y ^ { 2 } = 172.6$$ The population mean masses of these birds in countries \(X\) and \(Y\) are \(\mu _ { x } \mathrm {~kg}\) and \(\mu _ { y } \mathrm {~kg}\) respectively.
Test, at the \(5 \%\) significance level, the null hypothesis \(\mu _ { \mathrm { x } } = \mu _ { \mathrm { y } }\) against the alternative hypothesis \(\mu _ { \mathrm { x } } > \mu _ { \mathrm { y } }\). State your conclusion in the context of the question.

5 Dev owns a small company which produces bottles of juice. He uses two machines, \(X\) and \(Y\), to fill empty bottles with juice. Dev is investigating the volumes of juice in the bottles. He chooses a random sample of 35 bottles filled by machine \(X\) and a random sample of 60 bottles filled by machine \(Y\). The volumes of juice, \(x\) and \(y\) respectively, measured in suitable units, are summarised by $$\sum x = 30.8 , \quad \sum x ^ { 2 } = 29.0 , \quad \sum y = 62.4 , \quad \sum y ^ { 2 } = 76.8 .$$ Dev claims that the mean volume of juice in bottles filled by machine \(Y\) is greater than the mean volume of juice in bottles filled by machine \(X\). A test at the \(\alpha \%\) significance level suggests that there is sufficient evidence to support Dev's claim. Find the set of possible values of \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-10_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-11_2726_35_97_20}

8 Two machines, \(A\) and \(B\), produce metal rods. Machine \(B\) is new and it is required that its accuracy should be checked against that of machine \(A\). The observed variable is the length of a rod. Random samples of rods, 40 from machine \(A\) and 50 from machine \(B\), are taken and their lengths, \(x _ { A } \mathrm {~cm}\) and \(x _ { B } \mathrm {~cm}\), are measured. The results are summarised by $$\Sigma x _ { A } = 136.48 , \quad \Sigma x _ { B } = 176.35 , \quad \Sigma x _ { B } ^ { 2 } = 630.1940 .$$ The variance of the length of the rods produced by machine \(A\) is known to be \(0.0490 \mathrm {~cm} ^ { 2 }\). The mean lengths of the rods produced by the machines are denoted by \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively.

1. Test, at the \(5 \%\) significance level, the hypothesis \(\mu _ { B } > \mu _ { A }\).
2. Find the set of values of \(a\) for which the null hypothesis \(\mu _ { B } - \mu _ { A } = 0.025\) would not be rejected in favour of the alternative hypothesis \(\mu _ { B } - \mu _ { A } > 0.025\) at the \(a \%\) significance level.
3. For the test in part (i) to be valid,
   (a) state whether it is necessary to assume that the two population variances are equal,
   (b) state, giving a reason, whether it is necessary to assume that the lengths of rods are normally distributed.

3 An engineering company buys a certain type of component from two suppliers, A and B. It is important that, on the whole, the strengths of these components are the same from both suppliers. The company can measure the strengths in its laboratory. Random samples of seven components from supplier A and five from supplier B give the following strengths, in a convenient unit. The underlying distributions of strengths are assumed to be Normal for both suppliers, with variances 2.45 for supplier A and 1.40 for supplier B.

1. Test at the \(5 \%\) level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
2. Provide a two-sided 90\% confidence interval for the true mean difference.
3. Show that the test procedure used in part (i), with samples of sizes 7 and 5 and a \(5 \%\) significance level, leads to acceptance of the null hypothesis of equal means if \(- 1.556 < \bar { x } - \bar { y } < 1.556\), where \(\bar { x }\) and \(\bar { y }\) are the observed sample means from suppliers A and B . Hence find the probability of a Type II error for this test procedure if in fact the true mean strength from supplier A is 2.0 units more than that from supplier B.
4. A manager suggests that the Wilcoxon rank sum test should be used instead, comparing the median strengths for the samples of sizes 7 and 5 . Give one reason why this suggestion might be sensible and two why it might not.

5 Each person in a random sample of 15 men and 17 women from a university campus was asked how many days in a month they took exercise. The numbers of days for men and women, \(x _ { M }\) and \(x _ { W }\) respectively, are summarised by $$\Sigma x _ { M } = 221 , \quad \Sigma x _ { M } ^ { 2 } = 3992 , \quad \Sigma x _ { W } = 276 , \quad \Sigma x _ { W } ^ { 2 } = 5538 .$$

1. State conditions for the validity of a suitable test of the difference in the mean numbers of days for men and women on the campus.
2. Given that these conditions hold, carry out the test at the \(5 \%\) significance level.
3. If in fact the random sample was drawn entirely from the university Mathematics Department, state with a reason whether the validity of the test is in doubt.

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.

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.

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 \(n\).

3

1. At a waste disposal station, two methods for incinerating some of the rubbish are being compared. Of interest is the amount of particulates in the exhaust, which can be measured over the working day in a convenient unit of concentration. It is assumed that the underlying distributions of concentrations of particulates are Normal. It is also assumed that the underlying variances are equal. During a period of several months, measurements are made for method A on a random sample of 10 working days and for method B on a separate random sample of 7 working days, with results, in the convenient unit, as follows.

   Method A	124.8	136.4	116.6	129.1	140.7	120.2	124.6	127.5	111.8	130.3
   Method B	130.4	136.2	119.8	150.6	143.5	126.1	130.7

   Use a \(t\) test at the \(10 \%\) level of significance to examine whether either method is better in resulting, on the whole, in a lower concentration of particulates. State the null and alternative hypotheses under test.
2. The company's statistician criticises the design of the trial in part (i) on the grounds that it is not paired. Summarise the arguments the statistician will have used. A new trial is set up with a paired design, measuring the concentrations of particulates on a random sample of 9 paired occasions. The results are as follows.

   Pair	I	II	III	IV	V	VI	VII	VIII	IX
   Method A	119.6	127.6	141.3	139.5	141.3	124.1	116.6	136.2	128.8
   Method B	112.2	128.8	130.2	134.0	135.1	120.4	116.9	134.4	125.2

   Use a \(t\) test at the \(5 \%\) level of significance to examine the same hypotheses as in part (i). State the underlying distributional assumption that is needed in this case.
3. State the names of procedures that could be used in the situations of parts (i) and (ii) if the underlying distributional assumptions could not be made. What hypotheses would be under test?

A study was made of the acidity levels in farmland on opposite sides of an island. The levels were measured at six randomly chosen points on the eastern side and at five randomly chosen points on the western side. The values obtained, in suitable units, are denoted by \(x _ { E }\) and \(x _ { W }\) respectively. The sample means \(\bar { x } _ { E }\) and \(\bar { x } _ { W }\), and unbiased estimates of the two population variances, \(s _ { E } ^ { 2 }\) and \(s _ { W } ^ { 2 }\), are as follows. $$\bar { x } _ { E } = 5.035 , s _ { E } ^ { 2 } = 0.0231 , \bar { x } _ { W } = 4.782 , s _ { W } ^ { 2 } = 0.0195 .$$ The population means on the eastern and western sides are denoted by \(\mu _ { E }\) and \(\mu _ { W }\) respectively. State suitable hypotheses for a test for a difference between the mean acidity levels on the two sides of the island. Stating any required assumptions, obtain the rejection region for a test at the \(5 \%\) significance level of whether the mean acidity levels differ on the two sides of the island. Give the conclusion of the test. Find the largest value of \(a\) for which the samples above provide evidence at the \(5 \%\) significance level that \(\mu _ { E } - \mu _ { W } > a\).

9 Mr Lee asserts that boys are slower than girls at completing a particular mathematical puzzle. In order to test his assertion, a random sample of 40 boys and a random sample of 60 girls are selected from a large group of students who attempted the puzzle. The times taken by the boys, \(b\) minutes, and the times taken by the girls, \(g\) minutes, are summarised as follows. $$\Sigma b = 92.0 \quad \Sigma b ^ { 2 } = 216.5 \quad \Sigma g = 129.8 \quad \Sigma g ^ { 2 } = 288.8$$ Test at the \(2.5 \%\) significance level whether this evidence supports Mr Lee's assertion.

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.

8 Weekly expenses claimed by employees at two different branches, \(A\) and \(B\), of a large company are being compared. Expenses claimed by an employee at branch \(A\) and by an employee at branch \(B\) are denoted by \(\\) x\( and \)\\( y\) respectively. A random sample of 60 employees from branch \(A\) and a random sample of 50 employees from branch \(B\) give the following summarised data. $$\Sigma x = 6060 \quad \Sigma x ^ { 2 } = 626220 \quad \Sigma y = 4750 \quad \Sigma y ^ { 2 } = 464500$$ Using a \(2 \%\) significance level, test whether, on average, employees from branch \(A\) claim the same as employees from branch \(B\).

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 \(\alpha\).

9 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, \(x \mathrm {~kg}\), as follows. $$\begin{array} { l l l l l l l l } 1.2 & 1.4 & 0.8 & 2.1 & 1.8 & 2.6 & 1.5 & 2.0 \end{array}$$ Farmer \(Y\) chooses a random sample of 10 of his fish and summarises the masses, \(y \mathrm {~kg}\), 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 \(Y\).
[0pt] [10]

10 The times taken to run 400 metres by students at two large colleges \(P\) and \(Q\) are being compared. There is no evidence that the population variances are equal. The time taken by a student at college \(P\) and the time taken by a student at college \(Q\) are denoted by \(x\) seconds and \(y\) seconds respectively. A random sample of 50 students from college \(P\) and a random sample of 60 students from college \(Q\) give the following summarised data. $$\Sigma x = 2620 \quad \Sigma x ^ { 2 } = 138200 \quad \Sigma y = 3060 \quad \Sigma y ^ { 2 } = 157000$$

1. Using a 10\% significance level, test whether, on average, students from college \(P\) take longer to run 400 metres than students from college \(Q\).
2. Find a \(90 \%\) confidence interval for the difference in the mean times taken to run 400 metres by students from colleges \(P\) and \(Q\).

A farmer grows two different types of cherries, Type \(A\) and Type \(B\). He assumes that the masses of each type are normally distributed. He chooses a random sample of 8 cherries of Type \(A\). He finds that the sample mean mass is 15.1 g and that a \(95 \%\) confidence interval for the population mean mass, \(\mu \mathrm { g }\), is \(13.5 \leqslant \mu \leqslant 16.7\).

1. Find an unbiased estimate for the population variance of the masses of cherries of Type \(A\).
   The farmer now chooses a random sample of 6 cherries of Type \(B\) and records their masses as follows.
   12.2
   13.3
   13.9
   14.0
   15.4
   16.4
2. Test at the \(5 \%\) significance level whether the mean mass of cherries of Type \(B\) is less than the mean mass of cherries of Type \(A\). You should assume that the population variances for the two types of cherry are equal.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

A company produces packets of sweets. Two different machines, \(A\) and \(B\), are used to fill the packets. The manager decides to assess the performance of the two machines. He selects a random sample of 50 packets filled by machine \(A\) and a random sample of 60 packets filled by machine \(B\). The masses of sweets, \(x \mathrm {~kg}\), in packets filled by machine \(A\) and the masses of sweets, \(y \mathrm {~kg}\), in packets filled by machine \(B\) are summarised as follows. $$\Sigma x = 22.4 \quad \Sigma x ^ { 2 } = 10.1 \quad \Sigma y = 28.8 \quad \Sigma y ^ { 2 } = 16.3$$ A test at the \(\alpha \%\) significance level provides evidence that the mean mass of sweets in packets filled by machine \(A\) is less than the mean mass of sweets in packets filled by machine \(B\). Find the set of possible values of \(\alpha\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

A perfume manufacturer had one bottle-filling machine, but because of increased sales a second machine was obtained. In order to compare the performance of the two machines, a random sample of 50 bottles filled by the first machine and a random sample of 60 bottles filled by the second machine were checked. The volumes of the contents from the first machine, \(x _ { 1 } \mathrm { ml }\), and from the second machine, \(x _ { 2 } \mathrm { ml }\), are summarised by $$\Sigma x _ { 1 } = 1492.0 , \quad \Sigma x _ { 1 } ^ { 2 } = 44529.52 , \quad \Sigma x _ { 2 } = 1803.6 , \quad \Sigma x _ { 2 } ^ { 2 } = 54220.84 .$$ The volumes have distributions with means \(\mu _ { 1 } \mathrm { ml }\) and \(\mu _ { 2 } \mathrm { ml }\) for the first and second machines respectively. Test, at the \(2 \%\) significance level, whether \(\mu _ { 2 }\) is greater than \(\mu _ { 1 }\). Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { 2 } - \mu _ { 1 } > 0.1\).

9 The leaves from oak trees growing in two different areas \(A\) and \(B\) are being measured. The lengths, in cm , of a random sample of 7 oak leaves from area \(A\) are $$6.2 , \quad 8.3 , \quad 7.8 , \quad 9.3 , \quad 10.2 , \quad 8.4 , \quad 7.2$$ Assuming that the distribution is normal, find a 95\% confidence interval for the mean length of oak leaves from area \(A\). The lengths, in cm, of a random sample of 5 oak leaves from area \(B\) are $$5.9 , \quad 7.4 , \quad 6.8 , \quad 8.2 , \quad 8.7$$ Making suitable assumptions, which should be stated, test, at the \(5 \%\) significance level, whether the mean length of oak leaves from area \(A\) is greater than the mean length of oak leaves from area \(B\). [9]

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.