Normal distribution

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.

A company packs chickpeas into small bags and large bags.

The weight of a small bag of chickpeas is normally distributed with mean 500 g and standard deviation 5 g

A random sample of 3 small bags of chickpeas is taken.

Find the probability that the total weight of these 3 bags of chickpeas is between 1490 g and 1530 g

The weight of a large bag of chickpeas is normally distributed with mean 1020 g and standard deviation 20 g

One large bag and one small bag of chickpeas are chosen at random.

Calculate the probability that the weight of the large bag of chickpeas is at least 30 g more than twice the weight of the small bag of chickpeas. Show your working clearly.

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.

The weights of eggs, E grams, follow a normal distribution, N(60, 3²)

The weights of empty small boxes, S grams, follow a normal distribution, N(24, 1.8²)

The weights of empty large boxes, L grams, follow a normal distribution, N(40, 2.1²)

Small boxes of eggs contain 6 randomly selected eggs.

Large boxes of eggs contain 12 randomly selected eggs.

Find the probability that the total weight of a randomly selected small box of 6 eggs weighs less than 387 grams.

Find the probability that a randomly selected large box of 12 eggs weighs more than twice a randomly selected small box of 6 eggs.

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 random variable X ~ N(5, 0.4²) and the random variable Y ~ N(8, 0.1²)

X and Y are independent random variables.

A random sample of a independent observations is taken from the distribution of X and one observation is taken from the distribution of Y

The random variable W = X₁ + X₂ + X₃ +…+ X_a + bY and has the distribution N(169, 2²) Find the value of a and the value of b

The weights of a particular type of apple, A grams, and a particular type of orange, R grams, each follow independent normal distributions.

Find the distribution of

A + R

the total weight of 2 randomly selected apples.

A box contains 4 apples and 1 orange only. Jesse selects 2 pieces of fruit at random from the box.

Find the probability that the total weight of the 2 pieces of fruit exceeds 310 grams.

From a large number of apples and oranges, Celeste selects m apples and 1 orange at random. The random variable W is given by

where n is a positive integer.

Given that the middle 95% of the distribution of W lies between 1100.08 and 1499.92 grams,

find the value of m and the value of n

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.

Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm. The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm.

Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels.

One large panel and one small panel are selected at random.

Find the probability that the length of the large panel is more than times the length of the small panel.

Rosa needs 1000 cm of fencing. The large panels cost £80 each and the small panels cost £30 each. Rosa’s plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.

Calculate whether or not the expected cost of Rosa’s plan is cheaper than simply buying 14 small panels.

A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic.

The weights, W kg, of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg. The weights, kg, of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg. The random variables W and X are independent.

Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy.

The manufacturer packs n of these wooden toys and 2 n of these plastic toys into the same container. The maximum weight the container can hold is 252 kg.

The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.

Calculate the value of n.