F-test then t-test sequential

7. A psychologist gives a test to students from two different schools, \(A\) and \(B\). A group of 9 students is randomly selected from school \(A\) and given instructions on how to do the test.
A group of 7 students is randomly selected from school \(B\) and given the test without the instructions. The table shows the time taken, to the nearest second, to complete the test by the two groups. Stating your hypotheses clearly,

1. test at the \(10 \%\) significance level, whether or not the variance of the times taken to complete the test by students from school \(A\) is the same as the variance of the times taken to complete the test by students from school \(B\). (You may assume that times taken for each school are normally distributed.)
2. test at the \(5 \%\) significance level, whether or not the mean time taken to complete the test by students from school \(A\) is greater than the mean time taken to complete the test by students from school \(B\).
3. Why does the result to part (a) enable you to carry out the test in part (b)?
4. Give one factor that has not been taken into account in your analysis.

1. A large number of students are split into two groups \(A\) and \(B\). The students sit the same test but under different conditions. Group A has music playing in the room during the test, and group B has no music playing during the test. Small samples are then taken from each group and their marks recorded. The marks are normally distributed.

The marks are as follows:

1. Stating your hypotheses clearly, and using a \(10 \%\) level of significance, test whether or not there is evidence of a difference between the variances of the marks of the two groups.
2. State clearly an assumption you have made to enable you to carry out the test in part (a).
3. Use a two tailed test, with a \(5 \%\) level of significance, to determine if the playing of music during the test has made any difference in the mean marks of the two groups. State your hypotheses clearly.
4. Write down what you can conclude about the effect of music on a student's performance during the test.

1. A glue supplier claims that Goglue is stronger than Tackfast. A company is presently using Tackfast but agrees to change to Goglue if, at the 5\% significance level,

• the standard deviation of the force required for Goglue to fail is not greater than the standard deviation of the force required for Tackfast to fail and
• the mean force required for Goglue to fail is more than 4 newtons greater than the mean force for Tackfast to fail.

A series of trials is carried out, using Goglue and Tackfast, and the glues are tested to destruction. The force, \(x\) newtons, at which each glue fails is recorded. It can be assumed that the force at which each glue fails is normally distributed.

1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the standard deviation of the force required for Goglue to fail is greater than the standard deviation of the force required for the Tackfast to fail. State your hypotheses clearly. The supplier claims that the mean force required for its Goglue to fail is more than 4 newtons greater than the mean force required for Tackfast to fail.
2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the supplier's claim.
3. Show that, at the \(5 \%\) level of significance, the supplier's claim will be accepted if \(\bar { X } _ { G } - \bar { X } _ { T } > 4.55\), where \(\bar { X } _ { G }\) and \(\bar { X } _ { T }\) are the mean forces required for Goglue to fail and Tackfast to fail respectively. Later, it was found that an error had been made when recording the results for Goglue. This resulted in all the forces recorded for Goglue being 0.5 newtons more than they should have been. The results for Tackfast were correct.
4. Explain whether or not this information affects the decision about which glue the supplier decides to use.

5. An educational researcher is testing the effectiveness of a new method of teaching a topic in mathematics. A random sample of 10 children were taught by the new method and a second random sample of 9 children, of similar age and ability, were taught by the conventional method. At the end of the teaching, the same test was given to both groups of children. The marks obtained by the two groups are summarised in the table below.

1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, investigate whether or not
   1. the variance of the marks of children taught by the conventional method is greater than that of children taught by the new method,
   2. the mean score of children taught by the conventional method is lower than the mean score of those taught by the new method.
      [0pt] [In each case you should give full details of the calculation of the test statistics.]
2. State any assumptions you made in order to carry out these tests.
3. Find a 95\% confidence interval for the common variance of the marks of the two groups.

A grocer receives deliveries of cauliflowers from two different growers, \(A\) and \(B\). The grocer takes random samples of cauliflowers from those supplied by each grower. He measures the weight \(x\), in grams, of each cauliflower. The results are summarised in the table below. \includegraphics{figure_7}

1. Show, at the 10\% significance level, that the variances of the populations from which the samples are drawn can be assumed to be equal by testing the hypothesis H₀: \(\sigma_A^2 = \sigma_B^2\) against hypothesis H₁: \(\sigma_A^2 \neq \sigma_B^2\). (You may assume that the two samples come from normal populations.) [6]

The grocer believes that the mean weight of cauliflowers provided by \(B\) is at least 150 g more than the mean weight of cauliflowers provided by \(A\).

1. [(b)] Use a 5\% significance level to test the grocer's belief. [8]
2. Justify your choice of test. [2]

Two machines \(A\) and \(B\) produce the same type of component in a factory. The factory manager wishes to know whether the lengths, \(x\) cm, of the components produced by the two machines have the same mean. The manager took a random sample of components from each machine and the results are summarised in the table below. \includegraphics{figure_4} The lengths of components produced by the machines can be assumed to follow normal distributions.

1. Use a two tail test to show, at the 10\% significance level, that the variances of the lengths of components produced by each machine can be assumed to be equal. [4]
2. Showing your working clearly, find a 95\% confidence interval for \(\mu_A - \mu_B\), where \(\mu_A\) and \(\mu_B\) are the mean lengths of the populations of components produced by machine \(A\) and machine \(B\) respectively. [7]

There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm.

1. [(c)] State, giving your reason, whether or not the factory manager should be concerned. [2]

The lengths, \(x\) mm, of the forewings of a random sample of male and female adult butterflies are measured. The following statistics are obtained from the data. \includegraphics{figure_3}

1. Assuming the lengths of the forewings are normally distributed test, at the 10\% level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly. [7]
2. Stating your hypotheses clearly test, at the 5\% level of significance, whether the mean length of the forewings of the female butterflies is less than the mean length of the forewings of the male butterflies. [6]

A large number of students are split into two groups \(A\) and \(B\). The students sit the same test but under different conditions. Group A has music playing in the room during the test, and group B has no music playing during the test. Small samples are then taken from each group and their marks recorded. The marks are normally distributed. The marks are as follows: Sample from Group \(A\): 42, 40, 35, 37, 34, 43, 42, 44, 49 Sample from Group \(B\): 40, 44, 38, 47, 38, 37, 33

1. Stating your hypotheses clearly, and using a 10\% level of significance, test whether or not there is evidence of a difference between the variances of the marks of the two groups. [8]
2. State clearly an assumption you have made to enable you to carry out the test in part (a). [1]
3. Use a two tailed test, with a 5\% level of significance, to determine if the playing of music during the test has made any difference in the mean marks of the two groups. State your hypotheses clearly. [7]
4. Write down what you can conclude about the effect of music on a student's performance during the test. [1]

A farmer set up a trial to assess whether adding water to dry feed increases the milk yield of his cows. He randomly selected 22 cows. Thirteen of the cows were given dry feed and the other 9 cows were given the feed with water added. The milk yields, in litres per day, were recorded with the following results.

	Sample size	Mean	\(s^2\)
Dry feed	13	25.54	2.45
Feed with water added	9	27.94	1.02

You may assume that the milk yield from cows given the dry feed and the milk yield from cows given the feed with water added are from independent normal distributions.

1. Test, at the 10\% level of significance, whether or not the variances of the populations from which the samples are drawn are the same. State your hypotheses clearly. [5]
2. Calculate a 95\% confidence interval for the difference between the two mean milk yields. [7]
3. Explain the importance of the test in part (a) to the calculation in part (b). [2]