2.05a Hypothesis testing language: null, alternative, p-value, significance

2 A headteacher models the number of children who bring a 'healthy' packed lunch to school on any day by the distribution \(\mathrm { B } ( 150 , p )\). In the past, she has found that \(p = \frac { 1 } { 3 }\). Following the opening of a fast food outlet near the school, she wishes to test, at the \(1 \%\) significance level, whether the value of \(p\) has decreased.

1. State the null and alternative hypotheses for this test.
   On a randomly chosen day she notes the number, \(N\), of children who bring a 'healthy' packed lunch to school. She finds that \(N = 36\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \leqslant 36 ) = 0.0084\).
2. State, with a reason, the conclusion that the headteacher should draw from the test.
3. According to the model, what is the largest number of children who might bring a packed lunch to school?

5 The numbers of basketball courts in a random sample of 70 schools in South Mowland are summarised in the table.

1. Calculate unbiased estimates for the population mean and variance of the number of basketball courts per school in South Mowland.
   The mean number of basketball courts per school in North Mowland is 1.9 .
2. Test at the \(5 \%\) significance level whether the mean number of basketball courts per school in South Mowland is less than the mean for North Mowland.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (ii).

6 The number of accidents per month, \(X\), at a factory has a Poisson distribution. In the past the mean has been 1.1 accidents per month. Some new machinery is introduced and the management wish to test whether the mean has increased. They note the number of accidents in a randomly chosen month and carry out a hypothesis test at the 1\% significance level.

1. Show that the critical region for the test is \(X \geqslant 5\). Given that the number of accidents is 6 , carry out the test.
   Later they carry out a similar test, also at the \(1 \%\) significance level.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. Given that the mean is now 7.0 , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked.

1. It is suggested that the first 30 appointments on a Monday should be used for the test. Give a reason why this is not an appropriate sample.
   A suitable sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the 5\% significance level.
2. Explain why the test is one-tail and state suitable null and alternative hypotheses.
3. State what is meant by a Type I error in this context.
4. Use the binomial distribution to find the critical region, and find the probability of a Type I error.
5. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.

1 The time taken for Samuel to drive home from work is distributed with mean 46 minutes. Samuel discovers a different route and decides to test at the \(5 \%\) level whether the mean time has changed. He tries this route on a large number of different days chosen randomly and calculates the mean time.

1. State the null and alternative hypotheses for this test.
2. Samuel calculates the value of his test statistic \(z\) to be - 1.729 . What conclusion can he draw?

6 Pieces of metal discovered by people using metal detectors are found randomly in fields in a certain area at an average rate of 0.8 pieces per hectare. People using metal detectors in this area have a theory that ploughing the fields increases the average number of pieces of metal found per hectare. After ploughing, they tested this theory and found that a randomly chosen field of area 3 hectares yielded 5 pieces of metal.

1. Carry out the test at the \(10 \%\) level of significance.
2. What would your conclusion have been if you had tested at the \(5 \%\) level of significance? Jack decides that he will reject the null hypothesis that the average number is 0.8 pieces per hectare if he finds 4 or more pieces of metal in another ploughed field of area 3 hectares.
3. If the true mean after ploughing is 1.4 pieces per hectare, calculate the probability that Jack makes a Type II error.

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 Every month Susan enters a particular lottery. The lottery company states that the probability, \(p\), of winning a prize is 0.0017 each month. Susan thinks that the probability of winning is higher than this, and carries out a test based on her 12 lottery results in a one-year period. She accepts the null hypothesis \(p = 0.0017\) if she has no wins in the year and accepts the alternative hypothesis \(p > 0.0017\) if she wins a prize in at least one of the 12 months.

1. Find the probability of the test resulting in a Type I error.
2. If in fact the probability of winning a prize each month is 0.0024 , find the probability of the test resulting in a Type II error.
3. Use a suitable approximation, with \(p = 0.0024\), to find the probability that in a period of 10 years Susan wins a prize exactly twice.

4 The number of severe floods per year in a certain country over the last 100 years has followed a Poisson distribution with mean 1.8. Scientists suspect that global warming has now increased the mean. A hypothesis test, at the \(5 \%\) significance level, is to be carried out to test this suspicion. The number of severe floods, \(X\), that occur next year will be used for the test.

1. Show that the rejection region for the test is \(X > 4\).
2. Find the probability of making a Type II error if the mean number of severe floods is now actually 2.3.

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.

4 It is not known whether a certain coin is fair or biased. In order to perform a hypothesis test, Raj tosses the coin 10 times and counts the number of heads obtained. The probability of obtaining a head on any throw is denoted by \(p\).

1. The null hypothesis is \(p = 0.5\). Find the acceptance region for the test, given that the probability of a Type I error is to be at most 0.1 .
2. Calculate the probability of a Type II error in this test if the actual value of \(p\) is 0.7 .

6 It is claimed that a certain 6-sided die is biased so that it is more likely to show a six than if it was fair. In order to test this claim at the \(10 \%\) significance level, the die is thrown 10 times and the number of sixes is noted.

1. Given that the die shows a six on 3 of the 10 throws, carry out the test. On another occasion the same test is carried out again.
2. Find the probability of a Type I error.
3. Explain what is meant by a Type II error in this context.

7 In the past, the number of house sales completed per week by a building company has been modelled by a random variable which has the distribution \(\mathrm { Po } ( 0.8 )\). Following a publicity campaign, the builders hope that the mean number of sales per week will increase. In order to test at the \(5 \%\) significance level whether this is the case, the total number of sales during the first 3 weeks after the campaign is noted. It is assumed that a Poisson model is still appropriate.

1. Given that the total number of sales during the 3 weeks is 5 , carry out the test.
2. During the following 3 weeks the same test is carried out again, using the same significance level. Find the probability of a Type I error.
3. Explain what is meant by a Type I error in this context.
4. State what further information would be required in order to find the probability of a Type II error.

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 A cereal manufacturer claims that \(25 \%\) of cereal packets contain a free gift. Lola suspects that the true proportion is less than \(25 \%\). In order to test the manufacturer's claim at the \(5 \%\) significance level, she checks a random sample of 20 packets.

1. Find the critical region for the test.
2. Hence find the probability of a Type I error. Lola finds that 2 packets in her sample contain a free gift.
3. State, with a reason, the conclusion she should draw.

3 Joshi suspects that a certain die is biased so that the probability of showing a six is less than \(\frac { 1 } { 6 }\). He plans to throw the die 25 times and if it shows a six on fewer than 2 throws, he will conclude that the die is biased in this way.

1. Find the probability of a Type I error and state the significance level of the test. Joshi now decides to throw the die 100 times. It shows a six on 9 of these throws.
2. Calculate an approximate \(95 \%\) confidence interval for the probability of showing a six on one throw of this die.

7 The number of workers, \(X\), absent from a factory on a particular day has the distribution \(\mathrm { B } ( 80,0.01 )\).

1. Explain why it is appropriate to use a Poisson distribution as an approximating distribution for \(X\).
2. Use the Poisson distribution to find the probability that the number of workers absent during 12 randomly chosen days is more than 2 and less than 6 . Following a change in working conditions, the management wishes to test whether the mean number of workers absent per day has decreased.
3. During 10 randomly chosen days, there were a total of 2 workers absent. Use the Poisson distribution to carry out the test at the \(2 \%\) significance level.

6 At the last election, 70\% of people in Apoli supported the president. Luigi believes that the same proportion support the president now. Maria believes that the proportion who support the president now is \(35 \%\). In order to test who is right, they agree on a hypothesis test, taking Luigi's belief as the null hypothesis. They will ask 6 people from Apoli, chosen at random, and if more than 3 support the president they will accept Luigi's belief.

1. Calculate the probability of a Type I error.
2. If Maria's belief is true, calculate the probability of a Type II error.
3. In fact 2 of the 6 people say that they support the president. State which error, Type I or Type II, might be made. Explain your answer.

5 A fair six-sided die has faces numbered \(1,2,3,4,5,6\). The score on one throw is denoted by \(X\).

1. Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 35 } { 12 }\). Fayez has a six-sided die with faces numbered \(1,2,3,4,5,6\). He suspects that it is biased so that when it is thrown it is more likely to show a low number than a high number. In order to test his suspicion, he plans to throw the die 50 times. If the mean score is less than 3 he will conclude that the die is biased.
2. Find the probability of a Type I error.
3. With reference to this context, describe circumstances in which Fayez would make a Type II error.

6 The number of accidents on a certain road has a Poisson distribution with mean 3.1 per 12-week period.

1. Find the probability that there will be exactly 4 accidents during an 18-week period. Following the building of a new junction on this road, an officer wishes to determine whether the number of accidents per week has decreased. He chooses 15 weeks at random and notes the number of accidents. If there are fewer than 3 accidents altogether he will conclude that the number of accidents per week has decreased. He assumes that a Poisson distribution still applies.
2. Find the probability of a Type I error.
3. Given that the mean number of accidents per week is now 0.1 , find the probability of a Type II error.
4. Given that there were 2 accidents during the 15 weeks, explain why it is impossible for the officer to make a Type II error.

1 A researcher wishes to investigate whether the mean height of a certain type of plant in one region is different from the mean height of this type of plant everywhere else. He takes a large random sample of plants from the region and finds the sample mean. He calculates the value of the test statistic, \(z\), and finds that \(z = 1.91\).

1. Explain briefly why the researcher should use a two-tail test.
2. Carry out the test at the \(4 \%\) significance level.

5 It is known that when seeds of a certain type are planted, on average \(10 \%\) of the resulting plants reach a height of 1 metre. A gardener wishes to investigate whether a new fertiliser will increase this proportion. He plants a random sample of 18 seeds of this type, using the fertiliser, and notes how many of the resulting plants reach a height of 1 metre.

1. In fact 4 of the 18 plants reach a height of 1 metre. Carry out a hypothesis test at the \(8 \%\) significance level.
2. Explain which of the errors, Type I or Type II, might have been made in part (i). Later, the gardener plants another random sample of 18 seeds of this type, using the fertiliser, and again carries out a hypothesis test at the \(8 \%\) significance level.
3. Find the probability of a Type I error.

7 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked.

1. It is suggested that the first 30 appointments on a Monday should be used for the test. Give a reason why this is not an appropriate sample. A suitable sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the 5\% significance level.
2. Explain why the test is one-tail and state suitable null and alternative hypotheses.
3. State what is meant by a Type I error in this context.
4. Use the binomial distribution to find the critical region, and find the probability of a Type I error.
5. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.

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.