Simultaneous critical region and Type II error

5 Over a long period of time it is found that the time spent at cash withdrawal points follows a normal distribution with mean 2.1 minutes and standard deviation 0.9 minutes. A new system is tried out, to speed up the procedure. The null hypothesis is that the mean time spent is the same under the new system as previously. It is decided to reject the null hypothesis and accept that the new system is quicker if the mean withdrawal time from a random sample of 20 cash withdrawals is less than 1.7 minutes. Assume that, for the new system, the standard deviation is still 0.9 minutes, and the time spent still follows a normal distribution.

1. Calculate the probability of a Type I error.
2. If the mean withdrawal time under the new system is actually 1.5 minutes, calculate the probability of a Type II error.

5 The lectures in a mathematics department are scheduled to last 54 minutes, and the times of individual lectures may be assumed to have a normal distribution with mean \(\mu\) minutes and standard deviation 3.1 minutes. One of the students commented that, on average, the lectures seemed too short. To investigate this, the times for a random sample of 10 lectures were used to test the null hypothesis \(\mu = 54\) against the alternative hypothesis \(\mu < 54\) at the \(10 \%\) significance level.

1. Show that the null hypothesis is rejected in favour of the alternative hypothesis if \(\bar { x } < 52.74\), where \(\bar { x }\) minutes is the sample mean.
2. Find the probability of a Type II error given that the actual mean length of lectures is 51.5 minutes.

2 In summer the growth rate of grass in a lawn has a normal distribution with mean 3.2 cm per week and standard deviation 1.4 cm per week. A new type of grass is introduced which the manufacturer claims has a slower growth rate. A hypothesis test of this claim at the \(5 \%\) significance level was carried out using a random sample of 10 lawns that had the new grass. It may be assumed that the growth rate of the new grass has a normal distribution with standard deviation 1.4 cm per week.

1. Find the rejection region for the test.
2. The probability of making a Type II error when the actual value of the mean growth rate of the new grass is \(m \mathrm {~cm}\) per week is less than 0.5 . Use your answer to part (i) to write down an inequality for \(m\).

5 The management of a factory thinks that the mean time required to complete a particular task is 22 minutes. The times, in minutes, taken by employees to complete this task have a normal distribution with mean \(\mu\) and standard deviation 3.5. An employee claims that 22 minutes is not long enough for the task. In order to investigate this claim, the times for a random sample of 12 employees are used to test the null hypothesis \(\mu = 22\) against the alternative hypothesis \(\mu > 22\) at the \(5 \%\) significance level.

1. Show that the null hypothesis is rejected in favour of the alternative hypothesis if \(\bar { x } > 23.7\) (correct to 3 significant figures), where \(\bar { x }\) is the sample mean.
2. Find the probability of a Type II error given that the actual mean time is 25.8 minutes.

6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).

1. Given that \(n = 81\),
   1. find the critical region for the test, in terms of the sample mean \(\bar { W }\),
   2. find the probability that the test results in a Type II error when \(\mu = 50.2\).
   3. State how the probability of this Type II error would change if \(n\) were greater than 81 .

5 The time \(T\) seconds needed for a computer to be ready to use, from the moment it is switched on, is a normally distributed random variable with standard deviation 5 seconds. The specification of the computer says that the population mean time should be not more than 30 seconds.

1. A test is carried out, at the \(5 \%\) significance level, of whether the specification is being met, using the mean \(\bar { t }\) of a random sample of 10 times.
   1. Find the critical region for the test, in terms of \(\bar { t }\).
   2. Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.
   3. Because of system degradation and memory load, the population mean time \(\mu\) seconds increases with the number of months of use, \(m\). A formula for \(\mu\) in terms of \(m\) is \(\mu = 20 + 0.6 m\). Use this formula to find the value of \(m\) for which the probability that the test results in rejection of the null hypothesis is 0.5 .

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.

The random variable \(R\) has the distribution B(10, \(p\)). The null hypothesis H\(_0\): \(p = 0.7\) is to be tested against the alternative hypothesis H\(_1\): \(p < 0.7\), at a significance level of 5%.

1. Find the critical region for the test and the probability of making a Type I error. [3]
2. Given that \(p = 0.4\), find the probability that the test results in a Type II error. [3]
3. Given that \(p\) is equally likely to take the values 0.4 and 0.7, find the probability that the test results in a Type II error. [2]

Define, in terms of H\(_0\) and/or H\(_1\),

1. the size of a hypothesis test, [1]
2. the power of a hypothesis test. [1]

The probability of getting a head when a coin is tossed is denoted by \(p\). This coin is tossed 12 times in order to test the hypotheses H\(_0\): \(p = 0.5\) against H\(_1\): \(p \neq 0.5\), using a 5\% level of significance.

1. Find the largest critical region for this test, such that the probability in each tail is less than 2.5\%. [4]
2. Given that \(p = 0.4\)
   1. find the probability of a type II error when using this test,
   2. find the power of this test.
   [4]
3. Suggest two ways in which the power of the test can be increased. [2]