Two-sample t-test

3 At a factory, two production lines are in use for making steel rods. A critical dimension is the diameter of a rod. For the first production line, it is assumed from experience that the diameters are Normally distributed with standard deviation 1.2 mm . For the second production line, it is assumed from experience that the diameters are Normally distributed with standard deviation 1.4 mm . It is desired to test whether the mean diameters for the two production lines, \(\mu _ { 1 }\) and \(\mu _ { 2 }\), are equal. A random sample of 8 rods is taken from the first production line and, independently, a random sample of 10 rods is taken from the second production line.

1. Find the acceptance region for the customary test based on the Normal distribution for the null hypothesis \(\mu _ { 1 } = \mu _ { 2 }\), against the alternative hypothesis \(\mu _ { 1 } \neq \mu _ { 2 }\), at the \(5 \%\) level of significance.
2. The sample means are found to be 25.8 mm and 24.4 mm respectively. What is the result of the test? Provide a two-sided \(99 \%\) confidence interval for \(\mu _ { 1 } - \mu _ { 2 }\). The production lines are modified so that the diameters may be assumed to be of equal (but unknown) variance. However, they may no longer be Normally distributed. A two-sided test of the equality of the population medians is required, at the \(5 \%\) significance level.
3. The diameters in independent random samples of sizes 6 and 8 are as follows, in mm .

   First production line	25.9	25.8	25.3	24.7	24.4	25.4
   Second production line	23.8	25.6	24.0	23.5	24.1	24.5	24.3	25.1

   Use an appropriate procedure to carry out the test.

3 At an agricultural research station, trials are being carried out to compare a standard variety of tomato with one that has been genetically modified (GM). The trials are concerned with the mean weight of the tomatoes and also with the aesthetic appearance of the tomatoes.

1. 1. Tomatoes of the standard and GM varieties are grown under similar conditions. The tomatoes are weighed and the data are summarised as follows.

      Variety	Sample size	Sum of weights \(( \mathrm { g } )\)	Sum of squares of weights \(\left( \mathrm { g } ^ { 2 } \right)\)
      Standard	30	3218.3	349257
      GM	26	2954.1	338691

      Carry out a test, using the Normal distribution, to investigate whether there is evidence, at the 5\% level of significance, that the two varieties of tomato differ in mean weight. State one assumption required for this test to be valid.
   2. The data in part (i) could have been used to carry out a test for the equality of means based on the \(t\) distribution. State two additional assumptions required for this test to be valid. Discuss briefly which test would be preferable in this case.
2. In order to judge whether, on the whole, GM tomatoes have a better aesthetic appearance than standard tomatoes, a trial is carried out as follows. 10 of each variety are chosen and consumer panel is asked to arrange the 20 tomatoes in order according to their appearance.
   1. State two important features of the way in which this trial should be designed. Comment briefly on how reliable the evidence from the trial is likely to be.
   2. The order in which the consumer panel arranges the tomatoes is as follows. The tomato with best appearance is listed first. \(G\) and \(S\) denote GM and standard tomatoes respectively. $$\begin{array} { c c c c c c c c c c c c c c c c c c c c } G & G & G & S & G & G & G & S & G & S & S & S & G & G & S & G & S & S & S & S \end{array}$$ Carry out an appropriate test at the \(1 \%\) level of significance.

3 A large department in a university wished to compare the standards of literacy and numeracy of its students. A random sample of 24 students was taken and sub-divided, randomly, into two groups of 12 . The students in one group took a literacy assessment (scores denoted by \(x\) ); the students in the other group took a numeracy assessment (scores denoted by \(y\) ). The two assessments were designed to give the same distributions of scores when taken by random samples from the general population. The scores obtained by the students on the two assessments are shown in the table. $$\sum x = 598 \quad \sum x ^ { 2 } = 31196 \quad \sum y = 707 \quad \sum y ^ { 2 } = 43543$$

1. Carry out an appropriate \(t\) test, at the \(5 \%\) level of significance, to compare the standards of literacy and numeracy.
2. State the distributional assumptions required for the \(t\) test to be valid. Name the test that you would use if the assumptions required for the \(t\) test are thought not to hold. State the hypotheses for this new test. Explain, in general terms, which of the two tests is more powerful, and why. A statistician at the university looked at the data and commented that a paired sample design would have been better.
3. Explain how a paired sample design would be applied in this context, and how the data would be analysed. Explain also why it would be better than the design used.

7. Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml , are given in the table. One statistician suggests performing a two-sample \(t\)-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.

1. Stating your hypotheses clearly and using a \(5 \%\) significance level, carry out this test.
   (You may assume \(\bar { x } _ { A } = 26.125 , s _ { A } ^ { 2 } = 7.84 , \bar { x } _ { B } = 25 , s _ { B } ^ { 2 } = 4\) and \(\sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) ) Another statistician suggests analysing these data using a paired \(t\)-test.
2. Using a \(5 \%\) significance level, carry out this test.
3. State which of these two tests you consider to be more appropriate. Give a reason for your choice.

1. The times, \(x\) seconds, taken by the competitors in the 100 m freestyle events at a school swimming gala are recorded. The following statistics are obtained from the data.

Following the gala, a mother claims that girls are faster swimmers than boys. Assuming that the times taken by the competitors are two independent random samples from normal distributions,

1. test, at the \(10 \%\) level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly.
2. Stating your hypotheses clearly, test the mother's claim. Use a \(5 \%\) level of significance.