Tests for a variance

A factory produces bolts. The lengths of the bolts are normally distributed with mean μ mm and standard deviation 0.868 mm

A random sample of 15 of these bolts is taken and the mean length is 30.03 mm

Calculate a 90% confidence interval for μ

A suitable test, at the 10% level of significance, is carried out using these 15 bolts, to see whether or not there is evidence that the variance of the length of the bolts has increased.

Calculate the critical region for S²

The manager of the factory decides that, in future, he will check each month whether the machine making the bolts is working properly. He uses a 10% level of significance to test whether or not there is evidence that

• the mean length of the bolts has changed

• the variance of the length of the bolts has increased

The next month a random sample of 15 bolts is taken.

The mean length of these bolts is 30.06 mm and the standard deviation is 1.02 mm

With reference to your answers to part (a) and part (b), state whether or not there is any evidence that the machine is not working properly. Give reasons for your answer.

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 ml, dispensed in each bottle is recorded. The table shows the summary statistics for Machine A.

	Sample size	x	x2
Machine A	9	2268	571 700

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 ml²

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.

The concentration of an air pollutant is measured in micrograms/m³

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 H₀:σ_A² = σ_2B against the hypothesis H₁:σ_2A ≠ σ_2B is carried out using a 2% level of significance.

State a necessary assumption required to carry out the test.

Given that the assumption in part (a) holds,

find the set of values of s_B² that would lead to the null hypothesis being rejected,

find a 99% confidence interval for the variance of the concentration of the air pollutant at Site A.

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 cm, are summarised in the table below. The wingspans of Ringlet and Meadow Brown butterflies each follow normal distributions.

	Number of butterflies	w	w2
Ringlet	8	410	21 032
Meadow Brown	6	294	14 426

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)

Find the value of k

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.

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

[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.

A new employee, Kim, joins an existing employee, Jiang, to work in the quality control department of a company producing steel rods. Each day a random sample of rods is taken, their lengths measured and a 95% confidence interval for the mean length of the rods, in metres, is calculated. It is assumed that the lengths of the rods produced are normally distributed.

Kim took a random sample of 25 rods and used the t distribution to obtain a 95% confidence interval of (1.193, 1.367) for the mean length of the rods. Jiang commented that this interval was a little wider than usual and explained that they usually assume that the standard deviation does not change and can be taken as 0.175 metres.

Test, at the 10% level of significance, whether or not Kim’s sample suggests that the standard deviation is different from 0.175 metres. State your hypotheses clearly.

Using Kim’s sample and the normal distribution with a standard deviation of 0.175 metres,

find a 95% confidence interval for the mean length of the rods.

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 kg, produced by each plant is measured. The following statistics are obtained from the data.

	Number of plants	x	x2
A	25	194.7	1637.37
B	26	227.5	2031.19

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.

State an assumption you have made in order to carry out the test in part (a).