F-test for equality of variances

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.

2. The standard deviation of the length of a random sample of 8 fence posts produced by a timber yard was 8 mm . A second timber yard produced a random sample of 13 fence posts with a standard deviation of 14 mm .

1. Test, at the \(10 \%\) significance level, whether or not there is evidence that the lengths of fence posts produced by these timber yards differ in variability. State your hypotheses clearly.
2. State an assumption you have made in order to carry out the test in part (a).

2. The weights, in grams, of apples are assumed to follow a normal distribution. The weights of apples sold by a supermarket have variance \(\sigma _ { s } { } ^ { 2 }\). A random sample of 4 apples from the supermarket had weights $$\text { 114, 100, 119, } 123 .$$

1. Find a 95\% confidence interval for \(\sigma _ { s } ^ { 2 }\). The weights of apples sold on a market stall have variance \(\sigma _ { M } ^ { 2 }\). A second random sample of 7 apples was taken from the market stall. The sample variance \(s _ { M } ^ { 2 }\) of the apples was 318.8.
2. Stating your hypotheses clearly test, at the \(1 \%\) levcel of significnace, whether or not there is evidence that \(\sigma _ { M } ^ { 2 } > \sigma _ { s } ^ { 2 }\).

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. The sample variance of the lengths of a random sample of 9 paving slabs sold by a builders' merchant is \(36 \mathrm {~mm} ^ { 2 }\). The sample variance of the lengths of a random sample of 11 paving slabs sold by a second builders' merchant is \(225 \mathrm {~mm} ^ { 2 }\). Test at the \(10 \%\) significance level whether or not there is evidence that the lengths of paving slabs sold by these builders' merchants differ in variability. State your hypotheses clearly.
   (You may assume the lengths of paving slabs are normally distributed.)
   
2. A newspaper runs a daily Sudoku. A random sample of 10 people took the following times, in minutes, to complete the Sudoku.

\section*{\(\begin{array} { l l l l l l l l l l } 5.0 & 4.5 & 4.7 & 5.3 & 5.2 & 4.1 & 5.3 & 4.8 & 5.5 & 4.6 \end{array}\)} Given that the times to complete the Sudoku follow a normal distribution,

1. calculate a 95\% confidence interval for
   1. the mean,
   2. the variance,
      of the times taken by people to complete the Sudoku. The newspaper requires the average time needed to complete the Sudoku to be 5 minutes with a standard deviation of 0.7 minutes.
2. Comment on whether or not the Sudoku meets this requirement. Give a reason for your answer.

4. A company carries out an investigation into the strengths of rods from two different suppliers, Ardo and Bards. Independent random samples of rods were taken from each supplier and the force, \(x \mathrm { kN }\), needed to break each rod was recorded. The company wrote the results on a piece of paper but unfortunately spilt ink on it so some of the results can not be seen.
The paper with the results on is shown below. \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-08_435_1522_541_244}

1. 1. Use the data from Ardo to calculate an unbiased estimate, \(s _ { A } ^ { 2 }\), of the variance.
   2. Hence find an unbiased estimate, \(s _ { B } ^ { 2 }\), of the variance for the sample of 9 values from Bards.
2. Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not there is a difference in variability of strength between the rods from Ardo and the rods from Bards.
   (You may assume the two samples come from independent normal distributions.)
3. Use a \(5 \%\) level of significance to test whether the mean strength of rods from Bards is more than 0.9 kN greater than the mean strength of rods from Ardo.
   (6)

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.

3 Yin grows two varieties of potato, plant \(A\) and plant \(B\). A random sample of each variety of potato is taken and the yield, \(x \mathrm {~kg}\), produced by each plant is measured. The following statistics are obtained from the data.

1. Stating your hypotheses clearly, test, at the \(10 \%\) significance level, whether or not the variances of the yields of the two varieties of potato are the same.
2. State an assumption you have made in order to carry out the test in part (a).

1 Gina receives a large number of packages from two companies, \(A\) and \(B\). She believes that the variance of the weights of packages from company \(A\) is greater than the variance of the weights of packages from company \(B\). Gina takes a random sample of 7 packages from company \(A\) and an independent random sample of 10 packages from company \(B\). Her results are summarised below $$\bar { a } = 300 \quad \mathrm {~S} _ { a a } = 145496 \quad \bar { b } = 233.4 \quad \mathrm {~S} _ { b b } = 56364.4$$ [You may assume that the weights of packages from the two companies are normally distributed.]
Test Gina's belief. Use a \(5 \%\) level of significance and state your hypotheses clearly.

1. Elsa is collecting information on the wingspan of two different species of butterfly, Ringlet and Meadow Brown. She takes a random sample of each type of butterfly. The wingspans, \(w \mathrm {~cm}\), are summarised in the table below. The wingspans of Ringlet and Meadow Brown butterflies each follow normal distributions.

1. Test, at the \(2 \%\) level of significance, whether or not there is evidence that the variance of the wingspans of Ringlet butterflies is different from the variance of the wingspans of Meadow Brown butterflies. You should state your hypotheses clearly. The \(k \%\) confidence interval for the variance of the wingspans of Meadow Brown butterflies is (1.194, 48.54)
2. Find the value of \(k\)
3. Calculate a \(95 \%\) confidence interval for the difference between the mean wingspan of the Ringlet butterfly and the mean wingspan of the Meadow Brown butterfly.

1. The concentration of an air pollutant is measured in micrograms \(/ \mathrm { m } ^ { 3 }\)

Samples of air were taken at two different sites and the concentration of this particular air pollutant was recorded. For Site \(A\) the summary statistics are shown below. For Site \(B\) there were 9 samples of air taken.
A test of the hypothesis \(\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) against the hypothesis \(\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }\) is carried out using a \(2 \%\) level of significance.

1. State a necessary assumption required to carry out the test. Given that the assumption in part (a) holds,
2. find the set of values of \(s _ { B } ^ { 2 }\) that would lead to the null hypothesis being rejected,
3. find a 99\% confidence interval for the variance of the concentration of the air pollutant at Site A.

1. Two machines, \(A\) and \(B\), are used to fill bottles of water. The amount of water dispensed by each machine is normally distributed.

Samples are taken from each machine and the amount of water, \(x \mathrm { ml }\), dispensed in each bottle is recorded. The table shows the summary statistics for Machine \(A\).

1. Find a 95\% confidence interval for the variance of the amount of water dispensed in each bottle by Machine \(A\). For Machine \(B\), a random sample of 11 bottles is taken. The sample variance of the amount of water dispensed in bottles is \(12.7 \mathrm { ml } ^ { 2 }\)
2. Test, at the \(10 \%\) level of significance, whether there is evidence that the variances of the amounts of water dispensed in bottles by the two machines are different. You should state the hypotheses and the critical value used.

5. 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 proud parent 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 parent's claim. Use a \(5 \%\) level of significance.