Critical region determination

6 Last year, the mean time taken by students at a school to complete a certain test was 25 minutes. Akash believes that the mean time taken by this year's students was less than 25 minutes. In order to test this belief, he takes a large random sample of this year's students and he notes the time taken by each student. He carries out a test, at the \(2.5 \%\) significance level, for the population mean time, \(\mu\) minutes. Akash uses the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 25\).

1. Give a reason why Akash should use a one-tailed test.
   Akash finds that the value of the test statistic is \(z = - 2.02\).
2. Explain what conclusion he should draw.
   In a different one-tailed hypothesis test the \(z\)-value was found to be 2.14 .
3. Given that this value would lead to a rejection of the null hypothesis at the \(\alpha \%\) significance level, find the set of possible values of \(\alpha\).
   The population mean time taken by students at another school to complete a test last year was \(m\) minutes. Sorin carries out a one-tailed test to determine whether the population mean this year is less than \(m\), using a random sample of 100 students. He assumes that the population standard deviation of the times is 3.9 minutes. The sample mean is 24.8 minutes, and this result just leads to the rejection of the null hypothesis at the 5\% significance level.
4. Find the value of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

7 In the past, the mean time for Jenny's morning run was 28.2 minutes. She does some extra training and she wishes to test whether her mean time has been reduced. After the training Jenny takes a random sample of 40 morning runs. She decides that if the sample mean run time is less than 27 minutes she will conclude that the training has been effective. You may assume that, after the training, Jenny's run time has a standard deviation of 4.0 minutes.

1. State suitable null and alternative hypotheses for Jenny's test.
2. Find the probability that Jenny will make a Type I error.
3. Jenny found that the sample mean run time was 27.2 minutes. Explain briefly whether it is possible for her to make a Type I error or a Type II error or both.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The height \(H\), in metres, of mature trees of a certain variety is normally distributed with standard deviation 0.67. In order to test whether the population mean of \(H\) is greater than 4.23, the heights of a random sample of 200 trees are measured.

1. Write down suitable null and alternative hypotheses for the test.
   The sample mean height, \(\bar { h }\) metres, of the 200 trees is found and the test is carried out. The result of the test is to reject the null hypothesis at the 5\% significance level.
2. Find the set of possible values of \(\bar { h }\).
3. Ajit said, 'In (b) we had to assume that \(\bar { H }\) is normally distributed, so it was necessary to use the Central Limit Theorem.' Explain whether you agree with Ajit.

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.

4 The manufacturer of a tablet computer claims that the mean battery life is 11 hours. A consumer organisation wished to test whether the mean is actually greater than 11 hours. They invited a random sample of members to report the battery life of their tablets. They then calculated the sample mean. Unfortunately a fire destroyed the records of this test except for the following partial document. \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-2_467_593_1612_776} Given that the population of battery lives is normally distributed with standard deviation 1.6 hours, find the set of possible values of the sample size, \(n\).

9 It is desired to test whether the average amount of sleep obtained by school pupils in Year 11 is 8 hours, based on a random sample of size 64. The population standard deviation is 0.87 hours and the sample mean is denoted by \(\bar { H }\). The critical values for the test are \(\bar { H } = 7.72\) and \(\bar { H } = 8.28\).

1. State appropriate hypotheses for the test, explaining the meaning of any symbol you use.
2. Calculate the significance level of the test.
3. Explain what is meant by a Type I error in this context.
4. Given that in fact the average amount of sleep obtained by all pupils in Year 11 is 7.9 hours, find the probability that the test results in a Type II error. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

7 Three independent researchers, \(A , B\) and \(C\), carry out significance tests on the power consumption of a manufacturer's domestic heaters. The power consumption, \(X\) watts, is a normally distributed random variable with mean \(\mu\) and standard deviation 60. Each researcher tests the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 4000\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu > 4000\). Researcher \(A\) uses a sample of size 50 and a significance level of \(5 \%\).

1. Find the critical region for this test, giving your answer correct to 4 significant figures. In fact the value of \(\mu\) is 4020 .
2. Calculate the probability that Researcher \(A\) makes a Type II error.
3. Researcher \(B\) uses a sample bigger than 50 and a significance level of \(5 \%\). Explain whether the probability that Researcher \(B\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
4. Researcher \(C\) uses a sample of size 50 and a significance level bigger than \(5 \%\). Explain whether the probability that Researcher \(C\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
5. State with a reason whether it is necessary to use the Central Limit Theorem at any point in this question.

8 A random variable \(Y\) is normally distributed with mean \(\mu\) and variance 12.25. Two statisticians carry out significance tests of the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0\).

1. Statistician \(A\) uses the mean \(\bar { Y }\) of a sample of size 23, and the critical region for his test is \(\bar { Y } > 64.20\). Find the significance level for \(A\) 's test.
2. Statistician \(B\) uses the mean of a sample of size 50 and a significance level of \(5 \%\).
   (a) Find the critical region for \(B\) 's test.
   (b) Given that \(\mu = 65.0\), find the probability that \(B\) 's test results in a Type II error.
3. Given that, when \(\mu = 65.0\), the probability that \(A\) 's test results in a Type II error is 0.1365 , state with a reason which test is better.

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 \(n = 81\),
   (a) find the critical region for the test, in terms of the sample mean \(\bar { W }\),
   (b) find the probability that the test results in a Type II error when \(\mu = 50.2\).
2. 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.
   (a) Find the critical region for the test, in terms of \(\bar { t }\).
   (b) Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.
2. 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 .

8 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). A test is carried out, at the \(5 \%\) significance level, of \(\mathrm { H } _ { 0 } : \mu = 30\) against \(\mathrm { H } _ { 1 } : \mu > 30\), based on a random sample of size 18 .

1. Find the critical region for the test.
2. If \(\mu = 30\) and the outcome of the test is that \(\mathrm { H } _ { 0 }\) is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that \(\mathrm { H } _ { 0 }\) is rejected. It is known that the value of \(\mu\) remains the same throughout these 20 tests.
3. Find the probability that \(\mathrm { H } _ { 0 }\) is rejected at least 4 times if \(\mu = 30\). Hence state whether you think that \(\mu = 30\), giving a reason.
4. Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of \(\mu\), giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

4. A manufacturing company produces solar panels. The output of each solar panel is normally distributed with standard deviation 6 watts. It is thought that the mean output, \(\mu\), is 160 watts. A researcher believes that the mean output of the solar panels is greater than 160 watts. He writes down the output values of 5 randomly selected solar panels. He uses the data to carry out a hypothesis test at the \(5 \%\) level of significance. He tests \(\mathrm { H } _ { 0 } : \mu = 160\) against \(\mathrm { H } _ { 1 } : \mu > 160\) On reporting to his manager, the researcher can only find 4 of the output values. These are shown below $$\begin{array} { l l l l } 168.2 & 157.4 & 173.3 & 161.1 \end{array}$$ Given that the result of the hypothesis test is that there is significant evidence to reject \(\mathrm { H } _ { 0 }\) at the \(5 \%\) level of significance, calculate the minimum possible missing output value, \(\alpha\). Give your answer correct to 1 decimal place.

3. A clothes manufacturer wishes to find out if adult females have become taller on average since twenty years ago when their mean height was 5 ft 6 inches. Studies over time have shown that the standard deviation of the height of adult females has been fairly constant at 2.3 inches. The manager wishes to test if the mean height is now more than 5 ft 6 inches and takes a sample of 150 adult females.

1. Stating your hypotheses clearly, find the critical region for the mean height of the sample for a test at the \(5 \%\) level of significance. The total height of the females in the sample is 832 ft .
2. Carry out the test making your conclusion clear.

7. A machine produces bricks. The lengths, \(x \mathrm {~mm}\), of the bricks are distributed \(\mathrm { N } \left( \mu , 2 ^ { 2 } \right)\). At the start of each week a random sample of \(n\) bricks is taken to check the machine is working correctly.
A test is then carried out at the \(1 \%\) level of significance with $$\mathrm { H } _ { 0 } : \mu = 202 \text { and } \mathrm { H } _ { 1 } : \mu < 202$$

1. Find, in terms of \(n\), the critical region of the test. The probability of a type II error, when \(\mu = 200\), is less than 0.05
2. Find the minimum value of \(n\).

1. 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 \(\mu\) 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 $$\mathrm { H } _ { 0 } : \mu = 15 \quad \mathrm { H } _ { 1 } : \mu < 15$$

1. Write down the size of this test. Given that the actual value of \(\mu\) is 14.9
2. 1. calculate the minimum value of \(n\) such that the probability of a Type II error is less than 0.05
      Show your working clearly.
   2. 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.

1. A machine fills cartons with juice.

The amount of juice in a carton is normally distributed with mean \(\mu \mathrm { ml }\) and standard deviation 8 ml . A manager wants to test whether or not the amount of juice in the cartons, \(X \mathrm { ml }\), is less than 330 ml . The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice \(\bar { x } \mathrm { ml }\).

1. Using a \(5 \%\) level of significance, find the critical region of \(\bar { X }\) 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 \(\bar { y } \mathrm { ml }\) and uses a test with a critical region of $$\{ \bar { Y } < 328 \} \cup \{ \bar { Y } > 332 \}$$
2. Find P (Type I error) for the Director's test. When \(\mu = 325 \mathrm { ml }\)
3. find P (Type II error) for the test in part (a)

7 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). The mean of a random sample of 12 observations of \(X\) is denoted by \(\bar { X }\). A test is carried out at the \(1 \%\) significance level of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 80\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu < 80\). The test is summarised as follows: 'Reject \(\mathrm { H } _ { 0 }\) if \(\bar { X } < c\); otherwise do not reject \(\mathrm { H } _ { 0 } { } ^ { \prime }\).

1. Calculate the value of \(c\).
2. Assuming that \(\mu = 80\), state whether the conclusion of the test is correct, results in a Type I error, or results in a Type II error if:
   (a) \(\bar { X } = 74.0\),
   (b) \(\bar { X } = 75.0\).
3. Independent repetitions of the above test, using the value of \(c\) found in part (i), suggest that in fact the probability of rejecting the null hypothesis is 0.06 . Use this information to calculate the value of \(\mu\).

12 In the past, the time for Jeff's journey to work had mean 45.7 minutes and standard deviation 5.6 minutes. This year he is trying a new route. In order to test whether the new route has reduced his journey time, Jeff finds the mean time for a random sample of 30 journeys using the new route. He carries out a hypothesis test at the 2.5\% significance level. Jeff assumes that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.

1. State appropriate null and alternative hypotheses for the test.
2. Determine the rejection region for the test.