Confidence intervals

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.

Camilo grows two types of apple, green apples and red apples.

The standard deviation of the weights of green apples is known to be 3.5 grams.

A random sample of 80 green apples has a mean weight of 128 grams.

Find a 98% confidence interval for the mean weight of the population of green apples. Show your working clearly and give the confidence interval limits to 2 decimal places.

Camilo believes that the mean weight of the population of green apples is more than 10 grams greater than the mean weight of the population of red apples.

A random sample of n red apples has a mean weight of 117 grams.

The standard deviation of the weights of the red apples is known to be 4 grams.

A test of Camilo’s belief is carried out at the 5% level of significance.

State the null and alternative hypotheses for this test.

Find the smallest value of n for which the null hypothesis will be rejected.

Explain the relevance of the Central Limit Theorem in parts (a) and (c).

Given that n = 85, state the conclusion of the hypothesis test.

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.

A factory produces yellow tennis balls and white tennis balls. Independent samples, one of yellow tennis balls and one of white tennis balls, are taken. The table shows information about the weights of the yellow tennis balls, Y grams, and the weights of the white tennis balls, W grams.

Find a 95% confidence interval for the mean weight of yellow tennis balls.

Jamie claims that the mean weight of the population of yellow tennis balls is greater than the mean weight of the population of white tennis balls. A test of Jamie’s claim is carried out.

Specify the approximate distribution of – under the null hypothesis of the test.

Explain the relevance of the large sample sizes to your answer to part (i).

Complete the hypothesis test using a 5% level of significance. You should state your hypotheses and the value of your test statistic clearly.

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.

Calculate a 95% confidence interval for the mean weight of jam in a jar.

The labels on the jars state that the average contents weigh 440 grams.

State, giving a reason, whether or not Jemima should be concerned about the labels on her jars of jam.

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.

A machine is set to fill pots with yoghurt such that the mean weight of yoghurt in a pot is 505 grams.

To check that the machine is working properly, a random sample of 8 pots is selected. The weight of yoghurt, in grams, in each pot is as follows

508 510 500 500 498 503 508 505

Given that the weights of the yoghurt delivered by the machine follow a normal distribution with standard deviation 5.4 grams,

find a 95% confidence interval for the mean weight, grams, of yoghurt in a pot. Give your answers to 2 decimal places.

Comment on whether or not the machine is working properly, giving a reason for your answer.

State the probability that a 95% confidence interval for will not contain grams.

Without carrying out any further calculations, explain the changes, if any, that would need to be made in calculating the confidence interval in part (a) if the standard deviation was unknown. Give a reason for your answer. You may assume that the weights of the yoghurt delivered by the machine still follow a normal distribution.

A company manufactures bolts. The diameter of the bolts follows a normal distribution with a mean diameter of 5 mm.

Stan believes that the mean diameter of the bolts is less than 5 mm. He takes a random sample of 10 bolts and measures their diameters. He calculates some statistics but spills ink on his work before completing them. The only information he has left is as follows

99% confidence interval for the variance is = (0.01712, 0.23280)

Stating your hypotheses clearly, test, at the 5% level of significance, whether or not Stan’s belief is supported.