F-test two variances hypothesis

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 \(\sigma _ { M } ^ { 2 } > \sigma _ { s } ^ { 2 }\).

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)

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. 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.

A beach is divided into two areas \(A\) and \(B\). A random sample of pebbles is taken from each of the two areas and the length of each pebble is measured. A sample of size 26 is taken from area \(A\) and the unbiased estimate for the population variance is \(s_A^2 = 0.495 \text{ mm}^2\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s_B^2 = 1.04 \text{ mm}^2\).

1. Stating your hypotheses clearly test, at the 10\% significance level, whether or not there is a difference in variability of pebble length between area \(A\) and area \(B\). [5]
2. State the assumption you have made about the populations of pebble lengths in order to carry out this test. [1]

The random variable \(X\) has an \(F\)-distribution with 8 and 12 degrees of freedom. Find P\(\left(\frac{1}{5.67} < X < 2.85\right)\). [4]

The random variable \(X\) has a \(\chi^2\)-distribution with 9 degrees of freedom.

1. Find P(2.088 < \(X\) < 19.023). [3]

The random variable \(Y\) follows an \(F\)-distribution with 12 and 5 degrees of freedom.

1. [(b)] Find the upper and lower 5\% critical values for \(Y\). [3]

(Total 6 marks)

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. [5]
2. State an assumption you have made in order to carry out the test in part (a). [1]

(Total 6 marks)

A beach is divided into two areas \(A\) and \(B\). A random sample of pebbles is taken from each of the two areas and the length of each pebble is measured. A sample of size 26 is taken from area \(A\) and the unbiased estimate for the population variance is \(s_A^2 = 0.495\) mm\(^2\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s_B^2 = 1.04\) mm\(^2\).

1. Stating your hypotheses clearly test, at the 10\% significance level, whether or not there is a difference in variability of pebble length between area \(A\) and area \(B\). [5]
2. State the assumption you have made about the populations of pebble lengths in order to carry out the test. [1]

The sample variance of the lengths of a random sample of 9 paving slabs sold by a builders' merchant is 36 mm\(^2\). The sample variance of the lengths of a random sample of 11 paving slabs sold by a second builders' merchant is 225 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.) [5]