2.05c Significance levels: one-tail and two-tail

1 In a game, a ball is thrown and lands in one of 4 slots, labelled \(A , B , C\) and \(D\). Raju wishes to test whether the probability that the ball will land in slot \(A\) is \(\frac { 1 } { 4 }\).

1. State suitable null and alternative hypotheses for Raju's test.
   The ball is thrown 100 times and it lands in slot \(A 15\) times.
2. Use a suitable approximating distribution to carry out the test at the \(2 \%\) significance level.

2 Arvind uses an ordinary fair 6-sided die to play a game. He believes he has a system to predict the score when the die is thrown. Before each throw of the die, he writes down what he thinks the score will be. He claims that he can write the correct score more often than he would if he were just guessing. His friend Laxmi tests his claim by asking him to write down the score before each of 15 throws of the die. Arvind writes the correct score on exactly 5 out of 15 throws. Test Arvind's claim at the \(10 \%\) significance level.

4 The number of cars arriving at a certain road junction on a weekday morning has a Poisson distribution with mean 4.6 per minute. Traffic lights are installed at the junction and council officer wishes to test at the \(2 \%\) significance level whether there are now fewer cars arriving. He notes the number of cars arriving during a randomly chosen 2 -minute period.

1. State suitable null and alternative hypotheses for the test.
2. Find the critical region for the test.
   The officer notes that, during a randomly chosen 2 -minute period on a weekday morning, exactly 5 cars arrive at the junction.
3. Carry out the test.
4. State, with a reason, whether it is possible that a Type I error has been made in carrying out the test in part (c).
   The number of cars arriving at another junction on a weekday morning also has a Poisson distribution with mean 4.6 per minute.
5. Use a suitable approximating distribution to find the probability that more than 300 cars will arrive at this junction in an hour.

3 An architect wishes to investigate whether the buildings in a certain city are higher, on average, than buildings in other cities. He takes a large random sample of buildings from the city and finds the mean height of the buildings in the sample. He calculates the value of the test statistic, \(z\), and finds that \(z = 2.41\).

1. Explain briefly whether he should use a one-tail test or a two-tail test.
2. Carry out the test at the \(1 \%\) significance level.

6 It is known that \(8 \%\) of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.

1. He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5\% significance level.
2. Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).
   Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5\% significance level.
3. Find the probability of a Type I 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.

2 Harry has a five-sided spinner with sectors coloured blue, green, red, yellow and black. Harry thinks the spinner may be biased. He plans to carry out a hypothesis test with the following hypotheses. $$\begin{aligned} & \mathrm { H } _ { 0 } : \mathrm { P } ( \text { the spinner lands on blue } ) = \frac { 1 } { 5 } \\ & \mathrm { H } _ { 1 } : \mathrm { P } ( \text { the spinner lands on blue } ) \neq \frac { 1 } { 5 } \end{aligned}$$ Harry spins the spinner 300 times. It lands on blue on 45 spins.
Use a suitable approximation to carry out Harry's test at the \(5 \%\) significance level.

4 In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.5 and standard deviation 5.2. A new challenge is devised and it is expected that students will take, on average, less than 25.5 minutes to complete this challenge. A random sample of 40 students is chosen and their mean time for the new challenge is found to be 23.7 minutes.

1. Assuming that the standard deviation of the time for the new challenge is 5.2 minutes, test at the \(1 \%\) significance level whether the population mean time for the new challenge is less than 25.5 minutes.
2. State, with a reason, whether it is possible that a Type I error was made in the test in part (a).

6 The number of sports injuries per month at a certain college has a Poisson distribution. In the past the mean has been 1.1 injuries per month. The principal recently introduced new safety guidelines and she decides to test, at the \(2 \%\) significance level, whether the mean number of sports injuries has been reduced. She notes the number of sports injuries during a 6-month period.

1. Find the critical region for the test and state the probability of a Type I error.
2. State what is meant by a Type I error in this context.
3. During the 6 -month period there are a total of 2 sports injuries. Carry out the test.
4. Assuming that the mean remains 1.1 , calculate the probability that there will be fewer than 30 sports injuries during a 36-month period.

4 It is claimed that 1 in every 4 packets of certain biscuits contains a free gift. Marisa and André both suspect that the true proportion is less than 1 in 4.

1. Marisa chooses 20 packets at random. She decides that if fewer than 3 contain free gifts, she will conclude that the claim is not justified. Use a binomial distribution to find the probability of a Type I error.
2. André chooses 25 packets at random. He decides to carry out a significance test at the \(1 \%\) level, using a binomial distribution. Given that only 1 of the 25 packets contains a free gift, carry out the test.

7 In the past the number of accidents per month on a certain road was modelled by a random variable with distribution \(\operatorname { Po } ( 0.47 )\). After the introduction of speed restrictions, the government wished to test, at the 5\% significance level, whether the mean number of accidents had decreased. They noted the number of accidents during the next 12 months. It is assumed that accidents occur randomly and that a Poisson model is still appropriate.

1. Given that the total number of accidents during the 12 months was 2 , carry out the test.
2. Explain what is meant by a Type II error in this context.
   It is given that the mean number of accidents per month is now in fact 0.05 .
3. Using another random sample of 12 months the same test is carried out again, with the same significance level. Find the probability of a Type II error.

5 The time taken for a particular train journey is normally distributed. In the past, the time had mean 2.4 hours and standard deviation 0.3 hours. A new timetable is introduced and on 30 randomly chosen occasions the time for this journey is measured. The mean time for these 30 occasions is found to be 2.3 hours.

1. Stating any assumption(s), test, at the \(5 \%\) significance level, whether the mean time for this journey has changed.
2. A similar test at the \(5 \%\) significance level was carried out using the times from another randomly chosen 30 occasions.
   1. State the probability of a Type I error.
   2. State what is meant by a Type II error in this context.

3 When the council published a plan for a new road, only \(15 \%\) of local residents approved the plan. The council then published a revised plan and, out of a random sample of 300 local residents, 60 approved the revised plan. Is there evidence, at the \(2.5 \%\) significance level, that the proportion of local residents who approve the revised plan is greater than for the original plan?

5

1. The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by \(X\).
   1. State, with a justification, a suitable approximating distribution for \(X\), giving the values of any parameters.
   2. Use the approximating distribution to calculate \(\mathrm { P } ( X > 0 )\).
2. The percentage of people having a different medical condition is thought to be \(30 \%\). A researcher suspects that the true percentage is less than \(30 \%\). In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the \(2 \%\) significance level.

6 A machine is supposed to produce random digits. Bob thinks that the machine is not fair and that the probability of it producing the digit 0 is less than \(\frac { 1 } { 10 }\). In order to test his suspicion he notes the number of times the digit 0 occurs in 30 digits produced by the machine. He carries out a test at the \(10 \%\) significance level.

1. State suitable null and alternative hypotheses.
2. Find the rejection region for the test.
3. State the probability of a Type I error.
   It is now given that the machine actually produces a 0 once in every 40 digits, on average.
4. Find the probability of a Type II error.
5. Explain the meaning of a Type II error in this context.

4 At a certain airport 20\% of people take longer than an hour to check in. A new computer system is installed, and it is claimed that this will reduce the time to check in. It is decided to accept the claim if, from a random sample of 22 people, the number taking longer than an hour to check in is either 0 or 1 .

1. Calculate the significance level of the test.
2. State the probability that a Type I error occurs.
3. Calculate the probability that a Type II error occurs if the probability that a person takes longer than an hour to check in is now 0.09 .

4 In a certain city it is necessary to pass a driving test in order to be allowed to drive a car. The probability of passing the driving test at the first attempt is 0.36 on average. A particular driving instructor claims that the probability of his pupils passing at the first attempt is higher than 0.36 . A random sample of 8 of his pupils showed that 7 passed at the first attempt.

1. Carry out an appropriate hypothesis test to test the driving instructor's claim, using a significance level of \(5 \%\).
2. In fact, most of this random sample happened to be careful and sensible drivers. State which type of error in the hypothesis test (Type I or Type II) could have been made in these circumstances and find the probability of this type of error when a sample of size 8 is used for the test.

3 Metal bolts are produced in large numbers and have lengths which are normally distributed with mean 2.62 cm and standard deviation 0.30 cm .

1. Find the probability that a random sample of 45 bolts will have a mean length of more than 2.55 cm .
2. The machine making these bolts is given an annual service. This may change the mean length of bolts produced but does not change the standard deviation. To test whether the mean has changed, a random sample of 30 bolts is taken and their lengths noted. The sample mean length is \(m \mathrm {~cm}\). Find the set of values of \(m\) which result in rejection at the \(10 \%\) significance level of the hypothesis that no change in the mean length has occurred.

7 A hospital patient's white blood cell count has a Poisson distribution. Before undergoing treatment the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of the patient's white blood cell count is made, and is used to test at the \(10 \%\) significance level whether the mean white blood cell count has decreased.

1. State what is meant by a Type I error in the context of the question, and find the probability that the test results in a Type I error.
2. Given that the measured value of the white blood cell count after the treatment is 2 , carry out the test.
3. Find the probability of a Type II error if the mean white blood cell count after the treatment is actually 4.1.

1 At the 2009 election, \(\frac { 1 } { 3 }\) of the voters in Chington voted for the Citizens Party. One year later, a researcher questioned 20 randomly selected voters in Chington. Exactly 3 of these 20 voters said that if there were an election next week they would vote for the Citizens Party. Test at the \(2.5 \%\) significance level whether there is evidence of a decrease in support for the Citizens Party in Chington, since the 2009 election.

2 Dipak carries out a test, at the \(10 \%\) significance level, using a normal distribution. The null hypothesis is \(\mu = 35\) and the alternative hypothesis is \(\mu \neq 35\).

1. Is this a one-tail or a two-tail test? State briefly how you can tell. Dipak finds that the value of the test statistic is \(z = - 1.750\).
2. Explain what conclusion he should draw.
3. This result is significant at the \(\alpha \%\) level. Find the smallest possible value of \(\alpha\), correct to the nearest whole number.

4 At a power plant, the number of breakdowns per year has a Poisson distribution. In the past the mean number of breakdowns per year has been 4.8. Following some repairs, the management carry out a hypothesis test at the 5\% significance level to determine whether this mean has decreased. If there is at most 1 breakdown in the following year, they will conclude that the mean has decreased.

1. State what is meant by a Type I error in this context.
2. Find the probability of a Type I error.
3. Find the probability of a Type II error if the mean is now 0.9 breakdowns per year.

3 At an election in 2010, 15\% of voters in Bratfield voted for the Renewal Party. One year later, a researcher asked 30 randomly selected voters in Bratfield whether they would vote for the Renewal Party if there were an election next week. 2 of these 30 voters said that they would.

1. Use a binomial distribution to test, at the \(4 \%\) significance level, the null hypothesis that there has been no change in the support for the Renewal Party in Bratfield against the alternative hypothesis that there has been a decrease in support since the 2010 election.
2. (a) Explain why the conclusion in part (i) cannot involve a Type I error.
   (b) State the circumstances in which the conclusion in part (i) would involve a Type II error.

7 Previous records have shown that the number of cars entering Bampor on any day has mean 352 and variance 121.

1. Find the probability that the mean number of cars entering Bampor during a random sample of 200 days is more than 354 .
2. State, with a reason, whether it was necessary to assume that the number of cars entering Bampor on any day has a normal distribution in order to find the probability in part (i).
3. It is thought that the population mean may recently have changed. The number of cars entering Bampor during the day was recorded for each of a random sample of 50 days and the sample mean was found to be 356 . Assuming that the variance is unchanged, test at the \(5 \%\) significance level whether the population mean is still 352 .

2 A hockey player found that she scored a goal on \(82 \%\) of her penalty shots. After attending a coaching course, she scored a goal on 19 out of 20 penalty shots. Making an assumption that should be stated, test at the 10\% significance level whether she has improved.

6 Stephan is an athlete who competes in the high jump. In the past, Stephan has succeeded in \(90 \%\) of jumps at a certain height. He suspects that his standard has recently fallen and he decides to carry out a hypothesis test to find out whether he is right. If he succeeds in fewer than 17 of his next 20 jumps at this height, he will conclude that his standard has fallen.

1. Find the probability of a Type I error.
2. In fact Stephan succeeds in 18 of his next 20 jumps. Which of the errors, Type I or Type II, is possible? Explain your answer.