5.05c Hypothesis test: normal distribution for population mean

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).

3 The number of calls per day to an enquiry desk has a Poisson distribution. In the past the mean has been 5 . In order to test whether the mean has changed, the number of calls on a random sample of 10 days was recorded. The total number of calls was found to be 61 . Use an approximate distribution to test at the 10\% significance level whether the mean 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).

5 The mean breaking strength of cables made at a certain factory is supposed to be 5 tonnes. The quality control department wishes to test whether the mean breaking strength of cables made by a particular machine is actually less than it should be. They take a random sample of 60 cables. For each cable they find the breaking strength by gradually increasing the tension in the cable and noting the tension when the cable breaks.

1. Give a reason why it is necessary to take a sample rather then testing all the cables produced by the machine.
2. The mean breaking strength of the 60 cables in the sample is found to be 4.95 tonnes. Given that the population standard deviation of breaking strengths is 0.15 tonnes, test at the \(1 \%\) significance level whether the population mean breaking strength is less than it should be.
3. Explain whether it was necessary to use the Central Limit theorem in the solution to part (ii).

4 In the past, the time spent by customers in a certain shop had mean 12.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 13.5 minutes.

1. Assuming that the standard deviation remains at 4.2 minutes, test at the \(5 \%\) significance level whether the mean time spent by customers in the shop has changed.
2. Another random sample of 50 customers is chosen and a similar test at the \(5 \%\) significance level is carried out. State the probability of a Type I error.

4 At a certain company, computer faults occur randomly and at a constant mean rate. In the past this mean rate has been 2.1 per week. Following an update, the management wish to determine whether the mean rate has changed. During 20 randomly chosen weeks it is found that 54 computer faults occur. Use a suitable approximation to test at the \(5 \%\) significance level whether the mean rate has changed.

2 Past experience has shown that the heights of a certain variety of plant have mean 64.0 cm and standard deviation 3.8 cm . During a particularly hot summer, it was expected that the heights of plants of this variety would be less than usual. In order to test whether this was the case, a botanist recorded the heights of a random sample of 100 plants and found that the value of the sample mean was 63.3 cm . Stating a necessary assumption, carry out the test at the \(2.5 \%\) significance level.

3 Household incomes, in thousands of dollars, in a certain country are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The incomes of a random sample of 400 households are found and the results are summarised below. $$n = 400 \quad \Sigma x = 923 \quad \Sigma x ^ { 2 } = 3170$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. A random sample of 50 households in one particular region of the country is taken and the sample mean income, in thousands of dollars, is found to be 2.6 . Using your values from part (i), test at the \(5 \%\) significance level whether household incomes in this region are greater, on average, than in the country as a whole.

4 Last year the mean level of a certain pollutant in a river was found to be 0.034 grams per millilitre. This year the levels of pollutant, \(X\) grams per millilitre, were measured at a random sample of 200 locations in the river. The results are summarised below. $$n = 200 \quad \Sigma x = 6.7 \quad \Sigma x ^ { 2 } = 0.2312$$

1. Calculate unbiased estimates of the population mean and variance.
2. Test, at the \(10 \%\) significance level, whether the mean level of pollutant has changed.

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.

2

1. The time taken by a worker to complete a task was recorded for a random sample of 50 workers. The sample mean was 41.2 minutes and an unbiased estimate of the population variance was 32.6 minutes \({ } ^ { 2 }\). Find a \(95 \%\) confidence interval for the mean time taken to complete the task.
2. The probability that an \(\alpha \%\) confidence interval includes only values that are lower than the population mean is \(\frac { 1 } { 16 }\). Find the value of \(\alpha\).

3 Past experience has shown that the heights of a certain variety of rose bush have been normally distributed with mean 85.0 cm . A new fertiliser is used and it is hoped that this will increase the heights. In order to test whether this is the case, a botanist records the heights, \(x \mathrm {~cm}\), of a large random sample of \(n\) rose bushes and calculates that \(\bar { x } = 85.7\) and \(s = 4.8\), where \(\bar { x }\) is the sample mean and \(s ^ { 2 }\) is an unbiased estimate of the population variance. The botanist then carries out an appropriate hypothesis test.

1. The test statistic, \(z\), has a value of 1.786 correct to 3 decimal places. Calculate the value of \(n\).
2. Using this value of the test statistic, carry out the test at the \(5 \%\) significance level.

6 The number of injuries per month at a certain factory has a Poisson distribution. In the past the mean was 2.1 injuries per month. New safety procedures are put in place and the management wishes to use the next 3 months to test, at the \(2 \%\) significance level, whether there are now fewer injuries than before, on average.

1. Find the critical region for the test.
2. Find the probability of a Type I error.
3. During the next 3 months there are a total of 3 injuries. Carry out the test.
4. Assuming that the mean remains 2.1 , calculate an estimate of the probability that there will be fewer than 20 injuries during the next 12 months.

6 A survey taken last year showed that the mean number of computers per household in Branley was 1.66 . This year a random sample of 50 households in Branley answered a questionnaire with the following results.

1. Calculate unbiased estimates for the population mean and variance of the number of computers per household in Branley this year.
2. Test at the \(5 \%\) significance level whether the mean number of computers per household has changed since last year.
3. Explain whether it is possible that a Type I error may have been made in the test in part (ii).
4. State what is meant by a Type II error in the context of the test in part (ii), and give the set of values of the test statistic that could lead to a Type II error being made.

7 The masses, in grams, of apples from a certain farm have mean \(\mu\) and standard deviation 5.2. The farmer says that the value of \(\mu\) is 64.6. A quality control inspector claims that the value of \(\mu\) is actually less than 64.6. In order to test his claim he chooses a random sample of 100 apples from the farm.

1. The mean mass of the 100 apples is found to be 63.5 g . Carry out the test at the \(2.5 \%\) significance level.
2. Later another test of the same hypotheses at the \(2.5 \%\) significance level, with another random sample of 100 apples from the same farm, is carried out. Given that the value of \(\mu\) is in fact 62.7 , calculate 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.

4 A certain kind of firework is supposed to last for 30 seconds, on average, after it is lit. An inspector suspects that the fireworks actually last a shorter time than this, on average. He takes a random sample of 100 fireworks of this kind. Each firework in the sample is lit and the time it lasts is noted.

1. Give a reason why it is necessary to take a sample rather than testing all the fireworks of this kind.
   It is given that the population standard deviation of the times that fireworks of this kind last is 5 seconds.
2. The mean time lasted by the 100 fireworks in the sample is found to be 29 seconds. Test the inspector's suspicion at the \(1 \%\) significance level.
3. State with a reason whether the Central Limit theorem was needed in the solution to part (b).

7 In the past Laxmi's time, in minutes, for her journey to college had mean 32.5 and standard deviation 3.1. After a change in her route, Laxmi wishes to test whether the mean time has decreased. She notes her journey times for a random sample of 50 journeys and she finds that the sample mean is 31.8 minutes. You should assume that the standard deviation is unchanged.

1. Carry out a hypothesis test, at the \(8 \%\) significance level, of whether Laxmi's mean journey time has decreased.
   Later Laxmi carries out a similar test with the same hypotheses, at the \(8 \%\) significance level, using another random sample of size 50 .
2. Given that the population mean is now 31.5, 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.

5 In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.31 )\). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5 , carry out the test at the \(2.5 \%\) significance level.

7 A biologist wishes to test whether the mean concentration \(\mu\), in suitable units, of a certain pollutant in a river is below the permitted level of 0.5 . She measures the concentration, \(x\), of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. $$n = 50 \quad \Sigma x = 23.0 \quad \Sigma x ^ { 2 } = 13.02$$

1. Carry out a test at the \(5 \%\) significance level of the null hypothesis \(\mu = 0.5\) against the alternative hypothesis \(\mu < 0.5\).
   Later, a similar test is carried out at the \(5 \%\) significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.
2. Given that, in fact, the value of \(\mu\) is 0.4 , 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.

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 lengths, in centimetres, of worms of a certain kind are normally distributed with mean \(\mu\) and standard deviation 2.3. An article in a magazine states that the value of \(\mu\) is 12.7 . A scientist wishes to test whether this value is correct. He measures the lengths, \(x \mathrm {~cm}\), of a random sample of 50 worms of this kind and finds that \(\sum x = 597.1\). He plans to carry out a test, at the \(1 \%\) significance level, of whether the true value of \(\mu\) is different from 12.7 .

1. State, with a reason, whether he should use a one-tailed or a two-tailed test.
2. Carry out the test.

7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.

1. Calculate the probability of a Type I error.
2. Given that \(X = 2\), carry out the test. \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-10_2716_40_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-11_2716_29_107_22}
3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3 A factory owner models the number of employees who use the factory canteen on any day by the distribution \(\mathrm { B } ( 25 , p )\). In the past the value of \(p\) was 0.8 . A new menu is introduced in the canteen and the owner wants to test whether the value of \(p\) has increased. On a randomly chosen day he notes that the number of employees who use the canteen is 23 .

1. Use the binomial distribution to carry out the test at the \(10 \%\) significance level.
2. Given that there are 30 employees at the factory comment on the suitability of the owner's model. \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-04_2713_33_111_2016} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-05_2716_29_107_22}

7 The heights of one-year-old trees of a certain variety are known to have mean 2.3 m . A scientist believes that, on average, trees of this age and variety in her region are slightly taller than in other places. She plans to carry out a hypothesis test, at the \(2 \%\) significance level, in order to test her belief.

1. State the probability that she will make a Type I error.
   She takes a random sample of 100 such trees in her region and measures their heights, \(h \mathrm {~m}\). Her results are summarised below. $$n = 100 \quad \sum h = 238 \quad \sum h ^ { 2 } = 580$$
2. Carry out the test at the \(2 \%\) significance level. \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-10_2717_35_109_2012}
3. The scientist carries out the test correctly, but another scientist claims that she has made a Type II error. Comment on this claim.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.