Type I & Type II errors

A machine fills cartons with juice.

The amount of juice in a carton is normally distributed with mean µ ml and standard deviation 8 ml.

A manager wants to test whether or not the amount of juice in the cartons, X ml, is less than 330 ml. The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice ml.

Using a 5% level of significance, find the critical region of for this test. State your hypotheses clearly.

The Director is concerned about the machine filling the cartons with more than 330 ml of juice as well as less than 330 ml of juice. The Director takes a sample of 55 cartons, records the mean amount of juice ml and uses a test with a critical region of

Find P(Type I error) for the Director’s test.

When μ = 325 ml

find P(Type II error) for the test in part (a)

A machine fills bags with flour. The weight of flour delivered by the machine into a bag, X grams, is normally distributed with mean μ grams and standard deviation 30 grams. To check if there is any change to the mean weight of flour delivered by the machine into each bag, Olaf takes a random sample of 10 bags. The weight of flour, x grams, in each bag is recorded and = 1020

Test, at the 5% level of significance, H₀:μ = 1000 against H₁:μ 1000

Olaf decides to alter the test so that the hypotheses are H₀:μ = 1000 and H₁:μ 1000 but keeps the level of significance at 5% He takes a second sample of size n and finds the critical region, c

Find an equation for c in terms of n

When the true value of μ is 1020 grams, the probability of making a Type II error is 0.0050, to 2 significant figures.

Calculate the value of n and the value of c

Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red. Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.

The probability of Asha getting a red counter on any one draw is 0.07

Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.

Find the probability that Asha gets a red counter for the second time on her 9th draw.

The probability of Davinda getting a red counter on any one draw is p. Davinda draws counters until she gets n red counters. The random variable D is the number of counters Davinda draws.

Given that the mean and the standard deviation of D are 4400 and 660 respectively,

find the value of p.

Jerry believes that his bag contains a smaller proportion of red counters than Asha’s bag. To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable J to be the number of counters drawn up to and including the first red counter.

Stating your hypotheses clearly and using a 10% level of significance, find the critical region for this test.

Jerry gets a red counter for the first time on his 34th draw.

Giving a reason for your answer, state whether or not there is evidence that Jerry’s bag contains a smaller proportion of red counters than Asha’s bag.

Given that the probability of Jerry getting a red counter on any one draw is 0.011

show that the power of the test is 0.702 to 3 significant figures.

A manufacturer has a machine that produces lollipop sticks. The length of a lollipop stick produced by the machine is normally distributed with unknown mean μ and standard deviation 0.2

Farhan believes that the machine is not working properly and the mean length of the lollipop sticks has decreased. He takes a random sample of size n to test, at the 1% level of significance, the hypotheses

Write down the size of this test.

Given that the actual value of μ is 14.9

calculate the minimum value of n such that the probability of a Type II error is less than 0.05 Show your working clearly.

Farhan uses the same sample size, n, but now carries out the test at a 5% level of significance. Without doing any further calculations, state how this would affect the probability of a Type II error.

The number of customers entering Jeff’s supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.

Using this information,

find the probability that exactly 26 customers enter Jeff’s supermarket during a randomly selected 1-hour period one morning,

find the probability that at least 21 customers enter Jeff’s supermarket during a randomly selected 1-hour period one morning.

A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff’s supermarket over a randomly selected 40-minute period is found to be 10

Test, at the 5% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff’s supermarket. State your hypotheses clearly.

A further randomly selected 20-minute period is observed and the hypothesis test is repeated. Given that the true rate of customers entering Jeff’s supermarket is now 1 every 5 minutes,

calculate the probability of a Type II error.

A six-sided die has sides labelled 1, 2, 3, 4, 5 and 6

The random variable S represents the score when the die is rolled.

Alicia rolls the die 45 times and the mean score, , is calculated.

Assuming the die is fair and using a suitable approximation,

find, to 3 significant figures, the value of k such that P( < k) = 0.05

Explain the relevance of the Central Limit Theorem in part (a).

Alicia considers the following hypotheses:

H₀: The die is fair H: The die is not fair

Given that the true distribution of S has mean 4 and variance 3

find the power of this test.

Describe what would happen to the power of this test if Alicia were to increase the number of rolls of the die. Give a reason for your answer.