State meaning of Type I error

5 The time, in minutes, spent by customers at a particular gym has the distribution \(\mathrm { N } ( \mu , 38.2 )\). In the past the value of \(\mu\) has been 42.4. Following the installation of some new equipment the management wishes to test whether the value of \(\mu\) has changed.

1. State what is meant by a Type I error in this context.
2. The mean time for a sample of 20 customers is found to be 45.6 minutes. Test at the \(2.5 \%\) significance level whether the value of \(\mu\) has changed.

7 A researcher is investigating the actual lengths of time that patients spend with the doctor at their appointments. He plans to choose a sample of 12 appointments on a particular day.

1. Which of the following methods is preferable, and why?
   • Choose the first 12 appointments of the day.
   • Choose 12 appointments evenly spaced throughout the day.
   Appointments are scheduled to last 10 minutes. The actual lengths of time, in minutes, that patients spend with the doctor may be assumed to have a normal distribution with mean \(\mu\) and standard deviation 3.4. The researcher suspects that the actual time spent is more than 10 minutes on average. To test this suspicion, he recorded the actual times spent for a random sample of 12 appointments and carried out a hypothesis test at the 1\% significance level.
2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
3. Given that the total length of time spent for the 12 appointments was 147 minutes, carry out the test.
4. Give a reason why the Central Limit theorem was not needed in part (iii).

4 A study of a large sample of books by a particular author shows that the number of words per sentence can be modelled by a normal distribution with mean 21.2 and standard deviation 7.3. A researcher claims to have discovered a previously unknown book by this author. The mean length of 90 sentences chosen at random in this book is found to be 19.4 words.

1. Assuming the population standard deviation of sentence lengths in this book is also 7.3, test at the \(5 \%\) level of significance whether the mean sentence length is the same as the author's. State your null and alternative hypotheses.
2. State in words relating to the context of the test what is meant by a Type I error and state the probability of a Type I error in the test in part (i).

7 The number of cars caught speeding on a certain length of motorway is 7.2 per day, on average. Speed cameras are introduced and the results shown in the following table are those from a random selection of 40 days after this.

1. Calculate unbiased estimates of the population mean and variance of the number of cars per day caught speeding after the speed cameras were introduced.
2. Taking the null hypothesis \(\mathrm { H } _ { 0 }\) to be \(\mu = 7.2\), test at the \(5 \%\) level whether there is evidence that the introduction of speed cameras has resulted in a reduction in the number of cars caught speeding.
3. State what is meant by a Type I error in words relating to the context of the test in part (ii). Without further calculation, illustrate on a suitable diagram the region representing the probability of this Type I error.

4 In the past, the flight time, in hours, for a particular flight has had mean 6.20 and standard deviation 0.80 . Some new regulations are introduced. In order to test whether these new regulations have had any effect upon flight times, the mean flight time for a random sample of 40 of these flights is found.

1. State what is meant by a Type I error in this context.
2. The mean time for the sample of 40 flights is found to be 5.98 hours. Assuming that the standard deviation of flight times is still 0.80 hours, test at the \(5 \%\) significance level whether the population mean flight time has changed.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in your answer to part (ii).

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.

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 \(\mu = 58.5\) ?