Type I/II errors and power

6 The time taken by volunteers to complete a certain task is normally distributed. In the past the time, in minutes, has had mean 91.4 and standard deviation 6.4. A new, similar task is introduced and the times, \(t\) minutes, taken by a random sample of 6 volunteers to complete the new task are summarised by \(\Sigma t = 568.5\). Andrea plans to carry out a test, at the \(5 \%\) significance level, of whether the mean time for the new task is different from the mean time for the old task.

1. Give a reason why Andrea should use a two-tail test.
2. State the probability that a Type I error is made, and explain the meaning of a Type I error in this context.
   You may assume that the times taken for the new task are normally distributed.
3. Stating another necessary assumption, carry out the test.

6 Photographers often need to take many photographs of families until they find a photograph which everyone in the family likes. The number of photographs taken until obtaining one which everybody likes has mean 15.2. A new photographer claims that she can obtain a photograph which everybody likes with fewer photographs taken. To test at the \(10 \%\) level of significance whether this claim is justified, the numbers of photographs, \(x\), taken by the new photographer with a random sample of 60 families are recorded. The results are summarised by \(\Sigma x = 890\) and \(\Sigma x ^ { 2 } = 13780\).

1. Calculate unbiased estimates of the population mean and variance of the number of photographs taken by the new photographer.
2. State null and alternative hypotheses for the test, and state also the probability that the test results in a Type I error. Say what a Type I error means in the context of the question.
3. Carry out the test.

5 The masses of packets of cornflakes are normally distributed with standard deviation 11 g . A random sample of 20 packets was weighed and found to have a mean mass of 746 g .

1. Test at the \(4 \%\) significance level whether there is enough evidence to conclude that the population mean mass is less than 750 g .
2. Given that the population mean mass actually is 750 g , find the smallest possible sample size, \(n\), for which it is at least \(97 \%\) certain that the mean mass of the sample exceeds 745 g .

5 Records show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km . New driving regulations are introduced and in the next 20 weeks he drives a total of 21800 km .

1. Stating any assumption(s), test, at the \(1 \%\) significance level, whether his mean weekly driving distance has decreased.
2. A similar test at the \(1 \%\) significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context.
   [0pt] [2]

6 Parcels arriving at a certain office have weights \(W \mathrm {~kg}\), where the random variable \(W\) has mean \(\mu\) and standard deviation 0.2 . The value of \(\mu\) used to be 2.60 , but there is a suspicion that this may no longer be true. In order to test at the 5\% significance level whether the value of \(\mu\) has increased, a random sample of 75 parcels is chosen. You may assume that the standard deviation of \(W\) is unchanged.

1. The mean weight of the 75 parcels is found to be 2.64 kg . Carry out the test.
2. Later another test of the same hypotheses at the \(5 \%\) significance level, with another random sample of 75 parcels, is carried out. Given that the value of \(\mu\) is now 2.68 , calculate the probability of a Type II error.

7 In the past the time, in minutes, taken for a particular rail journey has been found to have mean 20.5 and standard deviation 1.2. Some new railway signals are installed. In order to test whether the mean time has decreased, a random sample of 100 times for this journey are noted. The sample mean is found to be 20.3 minutes. You should assume that the standard deviation is unchanged.

1. Carry out a significance test, at the \(4 \%\) level, of whether the population mean time has decreased. Later another significance test of the same hypotheses, using another random sample of size 100 , is carried out at the \(4 \%\) level.
2. Given that the population mean is now 20.1, find the probability of a Type II error.
3. State what is meant by a Type II error in this context.

8 The quantity, \(X\) milligrams per litre, of silicon dioxide in a certain brand of mineral water is a random variable with distribution \(\mathrm { N } \left( \mu , 5.6 ^ { 2 } \right)\).

1. A random sample of 80 observations of \(X\) has sample mean 100.7. Test, at the \(1 \%\) significance level, the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 102\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 102\).
2. The test is redesigned so as to meet the following conditions.
   • The hypotheses are \(\mathrm { H } _ { 0 } : \mu = 102\) and \(\mathrm { H } _ { 1 } : \mu < 102\).
   • The significance level is \(1 \%\).
   • The probability of making a Type II error when \(\mu = 100\) is to be (approximately) 0.05 .
   The sample size is \(n\), and the critical region is \(\bar { X } < c\), where \(\bar { X }\) denotes the sample mean.
   (a) Show that \(n\) and \(c\) satisfy (approximately) the equation \(102 - c = \frac { 13.0256 } { \sqrt { n } }\).
   (b) Find another equation satisfied by \(n\) and \(c\).
   (c) Hence find the values of \(n\) and \(c\).

3

1. Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic. A machine fills salt containers that will be sold in shops. The containers are supposed to contain 750 g of salt. The machine operates in such a way that the amount of salt delivered to each container is a Normally distributed random variable with standard deviation 20 g . The machine should be calibrated in such a way that the mean amount delivered, \(\mu\), is 750 g . Each hour, a random sample of 9 containers is taken from the previous hour's output and the sample mean amount of salt is determined. If this is between 735 g and 765 g , the previous hour's output is accepted. If not, the previous hour's output is rejected and the machine is recalibrated.
2. Find the probability of rejecting the previous hour's output if the machine is properly calibrated. Comment on your result.
3. Find the probability of accepting the previous hour's output if \(\mu = 725 \mathrm {~g}\). Comment on your result.
4. Obtain an expression for the operating characteristic of this testing procedure in terms of the cumulative distribution function \(\Phi ( z )\) of the standard Normal distribution. Evaluate the operating characteristic for the following values (in g) of \(\mu\) : 720, 730, 740, 750, 760, 770, 780.

3

1. Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.
2. A chemical manufacturer is endeavouring to reduce the amount of a certain impurity in one of its bulk products by improving the production process. The amount of impurity is measured in a convenient unit of concentration, and this is modelled by the Normally distributed random variable \(X\). In the old production process, the mean of \(X\), denoted by \(\mu\), was 63 and the standard deviation of \(X\) was 3.7. Experimental batches of the product are to be made using the new process, and it is desired to examine the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63\) and \(\mathrm { H } _ { 1 } : \mu < 63\) for the new process. Investigation of the variability in the new process has established that the standard deviation may be assumed unchanged. The usual Normal test based on \(\bar { X }\) is to be used, where \(\bar { X }\) is the mean of \(X\) over \(n\) experimental batches (regarded as a random sample), with a critical value \(c\) such that \(\mathrm { H } _ { 0 }\) is rejected if the value of \(\bar { X }\) is less than \(c\). The following criteria are set out.
   • If in fact \(\mu = 63\), the probability of concluding that \(\mu < 63\) must be only \(1 \%\).
   • If in fact \(\mu = 60\), the probability of concluding that \(\mu < 63\) must be \(90 \%\).
   Find \(c\) and the smallest value of \(n\) that is required. With these values, what is the power of the test if in fact \(\mu = 58.5\) ?

3 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). A hypothesis test is carried out of \(\mathrm { H } _ { 0 } : \mu = 20.0\) against \(\mathrm { H } _ { 1 } : \mu < 20.0\), at the \(1 \%\) level of significance, based on the mean of a sample of size 16. Given that in fact \(\mu = 15.0\), find the probability that the test results in a Type II error.

3

1. Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.
2. A market research organisation is designing a sample survey to investigate whether expenditure on everyday food items has increased in 2011 compared with 2010. For one of the populations being studied, the random variable \(X\) is used to model weekly expenditure, in \(\pounds\), on these items in 2011, where \(X\) is Normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). As the corresponding mean value in 2010 was 94 , the hypotheses to be examined are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 94 \\ & \mathrm { H } _ { 1 } : \mu > 94 \end{aligned}$$ By comparison with the corresponding 2010 value, \(\sigma ^ { 2 }\) is assumed to be 25 .
   The following criteria for the survey are laid down.
   • If in fact \(\mu = 94\), the probability of concluding that \(\mu > 94\) must be only \(2 \%\)
   • If in fact \(\mu = 97\), the probability of concluding that \(\mu > 94\) must be \(95 \%\)
   A random sample of size \(n\) is to be taken and the usual Normal test based on \(\bar { X }\) is to be used, with a critical value of \(c\) such that \(\mathrm { H } _ { 0 }\) is rejected if the value of \(\bar { X }\) exceeds \(c\). Find \(c\) and the smallest value of \(n\) that is required.
3. Sketch the power function of an ideal test for examining the hypotheses in part (ii).

6. A butter packing machine cuts butter into blocks. The weight of a block of butter is normally distributed with a mean weight of 250 g and a standard deviation of 4 g . A random sample of 15 blocks is taken to monitor any change in the mean weight of the blocks of butter.

1. Find the critical region of a suitable test using a \(2 \%\) level of significance.
   (3)
2. Assuming the mean weight of a block of butter has increased to 254 g , find the probability of a Type II error.

1. A machine makes posts. The length of a post is normally distributed with unknown mean \(\mu\) and standard deviation 4 cm .

A random sample of size \(n\) is taken to test, at the \(5 \%\) significance level, the hypotheses $$\mathrm { H } _ { 0 } : \mu = 150 \quad \mathrm { H } _ { 1 } : \mu > 150$$

1. State the probability of a Type I error for this test. The manufacturer requires the probability of a Type II error to be less than 0.1 when the actual value of \(\mu\) is 152
2. Calculate the minimum value of \(n\).

2 Amy takes a sample of size 50 from a normal distribution with mean \(\mu\) and variance 16 She conducts a hypothesis test with hypotheses: $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 52 \\ & \mathrm { H } _ { 1 } : \mu > 52 \end{aligned}$$ She rejects the null hypothesis if her sample has a mean greater than 53
The actual population mean is 53.5
Find the probability that Amy makes a Type II error.
Circle your answer. \(0.4 \% 3.9 \% 18.9 \% 15.0 \%\)

2 Jamie is conducting a hypothesis test on a random variable which has a normal distribution with standard deviation 1 The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 5 \\ & \mathrm { H } _ { 1 } : \mu > 5 \end{aligned}$$ He takes a random sample of size 4
The mean of his sample is 6
He uses a 5\% level of significance.
Before Jamie conducted the test, what was the probability that he would make a Type I error? Circle your answer.
[0pt] [1 mark] \(0.0228 \quad 0.0456 \quad 0.0500 \quad 0.1587\)