5.05c Hypothesis test: normal distribution for population mean

2 A random sample of 40 observations of a random variable \(X\) and a random sample of 50 observations of a random variable \(Y\) are taken. The resulting values for the sample means, \(\bar { x }\) and \(\bar { y }\), and the unbiased estimates, \(\mathrm { s } _ { \mathrm { x } } ^ { 2 }\) and \(\mathrm { s } _ { \mathrm { y } } ^ { 2 }\), for the population variances are as follows. $$\bar { x } = 24.4 \quad \bar { y } = 17.2 \quad s _ { x } ^ { 2 } = 10.2 \quad s _ { y } ^ { 2 } = 11.1$$ Find a \(90 \%\) confidence interval for the difference between the population means of \(X\) and \(Y\).

5 Students at two colleges, \(A\) and \(B\), are competing in a computer games challenge.

1. The time taken for a randomly chosen student from college \(A\) to complete the challenge has a normal distribution with mean \(\mu\) minutes. The times taken, \(x\) minutes, are recorded for a random sample of 10 students chosen from college \(A\). The results are summarised as follows. $$\sum x = 828 \quad \sum x ^ { 2 } = 68622$$ A test is carried out on the data at the \(5 \%\) significance level and the result supports the claim that \(\mu > k\). Find the greatest possible value of \(k\).
2. A random sample of 8 students is chosen from college \(B\). Their times to complete the same challenge give a sample mean of 79.8 minutes and an unbiased variance estimate of 9.966 minutes \({ } ^ { 2 }\). Use a 2 -sample test at the \(5 \%\) significance level to test whether the mean time for students at college \(B\) to complete the challenge is the same as the mean time for students at college \(A\) to complete the challenge. You should assume that the two distributions are normal and have the same population variance.

1 A random sample of 7 observations of a variable \(X\) are as follows. $$\begin{array} { l l l l l l l } 8.26 & 7.78 & 7.92 & 8.04 & 8.27 & 7.95 & 8.34 \end{array}$$ The population mean of \(X\) is \(\mu\).

1. Test, at the \(10 \%\) significance level, the null hypothesis \(\mu = 8.22\) against the alternative hypothesis \(\mu < 8.22\).
2. State an assumption necessary for the test in part (a) to be valid.

4 A scientist is investigating the lengths of the leaves of birch trees in different regions. He takes a random sample of 50 leaves from birch trees in region \(A\) and a random sample of 60 leaves from birch trees in region \(B\). He records their lengths in \(\mathrm { cm } , x\) and \(y\), respectively. His results are summarised as follows. $$\sum x = 282 \quad \sum x ^ { 2 } = 1596 \quad \sum y = 328 \quad \sum y ^ { 2 } = 1808$$ The population mean lengths of leaves from birch trees in regions \(A\) and \(B\) are \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively. Carry out a test at the \(5 \%\) significance level to test the null hypothesis \(\mu _ { \mathrm { A } } = \mu _ { \mathrm { B } }\) against the alternative hypothesis \(\mu _ { \mathrm { A } } \neq \mu _ { \mathrm { B } }\).

4 A company has two different machines, \(X\) and \(Y\), each of which fills empty cups with coffee. The manager is investigating the volumes of coffee, \(x\) and \(y\), measured in appropriate units, in the cups filled by machines \(X\) and \(Y\) respectively. She chooses a random sample of 50 cups filled by machine \(X\) and a random sample of 40 cups filled by machine \(Y\). The volumes are summarised as follows. $$\sum x = 15.2 \quad \sum x ^ { 2 } = 5.1 \quad \sum y = 13.4 \quad \sum y ^ { 2 } = 4.8$$ The manager claims that there is no difference between the mean volume of coffee in cups filled by machine \(X\) and the mean volume of coffee in cups filled by machine \(Y\). Test the manager's claim at the \(10 \%\) significance level.

5 A large number of children are competing in a throwing competition. The distances, in metres, thrown by a random sample of 8 children are as follows. \(\begin{array} { l l l l l l l l } 19.8 & 22.1 & 24.4 & 21.5 & 20.8 & 26.3 & 23.7 & 25.0 \end{array}\)

1. Assuming that distances are normally distributed, test, at the \(5 \%\) significance level, whether the population mean distance thrown is more than 22.0 metres.
2. Find a 95\% confidence interval for the population mean distance thrown.

3 At factory \(A\) the mean number of accidents per year is 32 . At factory \(B\) the records of numbers of accidents before 2018 have been lost, but the number of accidents during 2018 was 21. It is known that the number of accidents per year can be well modelled by a Poisson distribution. Use an approximating distribution to test at the \(2 \%\) significance level whether the mean number of accidents at factory \(B\) is less than at factory \(A\).

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.

5 The distance driven in a week by a long-distance lorry driver is a normally distributed random variable with mean 1850 km and standard deviation 117 km .

1. Find the probability that in a random sample of 26 weeks his average distance driven per week is more than 1800 km .
2. New driving regulations are introduced and in a random sample of 26 weeks after their introduction the lorry driver drives a total of 47658 km . Assuming the standard deviation remains unchanged, test at the \(10 \%\) level whether his mean weekly driving distance has changed.

7 In a research laboratory where plants are studied, the probability of a certain type of plant surviving was 0.35 . The laboratory manager changed the growing conditions and wished to test whether the probability of a plant surviving had increased.

1. The plants were grown in rows, and when the manager requested a random sample of 8 plants to be taken, the technician took all 8 plants from the front row. Explain what was wrong with the technician's sample.
2. A suitable sample of 8 plants was taken and 4 of these 8 plants survived. State whether the manager's test is one-tailed or two-tailed and also state the null and alternative hypotheses. Using a \(5 \%\) significance level, find the critical region and carry out the test.
3. State the meaning of a Type II error in the context of the test in part (ii).
4. Find the probability of a Type II error for the test in part (ii) if the probability of a plant surviving is now 0.4.

3 Flies stick to wet paint at random points. The average number of flies is 2 per square metre. A wall with area \(22 \mathrm {~m} ^ { 2 }\) is painted with a new type of paint which the manufacturer claims is fly-repellent. It is found that 27 flies stick to this wall. Use a suitable approximation to test the manufacturer's claim at the \(1 \%\) significance level. Take the null hypothesis to be \(\mu = 44\), where \(\mu\) is the population mean.

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.

5 The masses of packets of cornflakes are normally distributed with standard deviation 11 g . A random sample of 20 packets was weighed and found to have a mean mass of 746 g .

1. Test at the \(4 \%\) significance level whether there is enough evidence to conclude that the population mean mass is less than 750 g .
2. Given that the population mean mass actually is 750 g , find the smallest possible sample size, \(n\), for which it is at least \(97 \%\) certain that the mean mass of the sample exceeds 745 g .

5 The marks of candidates in Mathematics and English in 2009 were represented by the independent random variables \(X\) and \(Y\) with distributions \(\mathrm { N } \left( 28,5.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 52,12.4 ^ { 2 } \right)\) respectively. Each candidate's marks were combined to give a final mark \(F\), where \(F = X + \frac { 1 } { 2 } Y\).

1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
2. The final marks of a random sample of 10 candidates from Grinford in 2009 had a mean of 49. Test at the 5\% significance level whether this result suggests that the mean final mark of all candidates from Grinford in 2009 was lower than elsewhere.

3 The masses of sweets produced by a machine are normally distributed with mean \(\mu\) grams and standard deviation 1.0 grams. A random sample of 65 sweets produced by the machine has a mean mass of 29.6 grams.

1. Find a \(99 \%\) confidence interval for \(\mu\). The manufacturer claims that the machine produces sweets with a mean mass of 30 grams.
2. Use the confidence interval found in part (i) to draw a conclusion about this claim.
3. Another random sample of 65 sweets produced by the machine is taken. This sample gives a \(99 \%\) confidence interval that leads to a different conclusion from that found in part (ii). Assuming that the value of \(\mu\) has not changed, explain how this can be possible.

6 A clinic monitors the amount, \(X\) milligrams per litre, of a certain chemical in the blood stream of patients. For patients who are taking drug \(A\), it has been found that the mean value of \(X\) is 0.336 . A random sample of 100 patients taking a new drug, \(B\), was selected and the values of \(X\) were found. The results are summarised below. $$n = 100 , \quad \Sigma x = 43.5 , \quad \Sigma x ^ { 2 } = 31.56 .$$

1. Test at the \(1 \%\) significance level whether the mean amount of the chemical in the blood stream of patients taking drug \(B\) is different from that of patients taking drug \(A\).
2. For the test to be valid, is it necessary to assume a normal distribution for the amount of chemical in the blood stream of patients taking drug \(B\) ? Justify your answer.

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.

2 The heights of a certain type of plant have a normal distribution. When the plants are grown without fertilizer, the population mean and standard deviation are 24.0 cm and 4.8 cm respectively. A gardener wishes to test, at the \(2 \%\) significance level, whether Hiergro fertilizer will increase the mean height. He treats 150 randomly chosen plants with Hiergro and finds that their mean height is 25.0 cm . Assuming that the standard deviation of the heights of plants treated with Hiergro is still 4.8 cm , carry out the test.

2

1. A random variable \(X\) has mean \(\mu\) and variance \(\sigma ^ { 2 }\). The mean of a random sample of \(n\) values of \(X\) is denoted by \(\bar { X }\). Give expressions for \(\mathrm { E } ( \bar { X } )\) and \(\operatorname { Var } ( \bar { X } )\).
2. The heights, in centimetres, of adult males in Brancot are normally distributed with mean 177.8 and standard deviation 6.1. Find the probability that the mean height of a random sample of 12 adult males from Brancot is less than 176 cm .
3. State, with a reason, whether it was necessary to use the Central Limit Theorem in the calculation in part (ii).

5 It is claimed that, on average, people following the Losefast diet will lose more than 2 kg per month. The weight losses, \(x\) kilograms per month, of a random sample of 200 people following the Losefast diet were recorded and summarised as follows. $$n = 200 \quad \Sigma x = 460 \quad \Sigma x ^ { 2 } = 1636$$

1. Calculate unbiased estimates of the population mean and variance.
2. Test the claim at the \(1 \%\) significance level.

3 Following a change in flight schedules, an airline pilot wished to test whether the mean distance that he flies in a week has changed. He noted the distances, \(x \mathrm {~km}\), that he flew in 50 randomly chosen weeks and summarised the results as follows. $$n = 50 \quad \Sigma x = 143300 \quad \Sigma x ^ { 2 } = 410900000$$

1. Calculate unbiased estimates of the population mean and variance.
2. In the past, the mean distance that he flew in a week was 2850 km . Test, at the \(5 \%\) significance level, whether the mean distance has changed.

2 A traffic officer notes the speeds of vehicles as they pass a certain point. In the past the mean of these speeds has been \(62.3 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the standard deviation has been \(10.4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). A speed limit is introduced, and following this, the mean of the speeds of 75 randomly chosen vehicles passing the point is found to be \(59.9 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. Making an assumption that should be stated, test at the \(2 \%\) significance level whether the mean speed has decreased since the introduction of the speed limit.
2. Explain whether it was necessary to use the Central Limit theorem in part (i).

5 The number of hours that Mrs Hughes spends on her business in a week is normally distributed with mean \(\mu\) and standard deviation 4.8. In the past the value of \(\mu\) has been 49.5.

1. Assuming that \(\mu\) is still equal to 49.5 , find the probability that in a random sample of 40 weeks the mean time spent on her business in a week is more than 50.3 hours. Following a change in her arrangements, Mrs Hughes wishes to test whether \(\mu\) has decreased. She chooses a random sample of 40 weeks and notes that the total number of hours she spent on her business during these weeks is 1920.
2. (a) Explain why a one-tail test is appropriate.
   (b) Carry out the test at the 6\% significance level.
   (c) Explain whether it was necessary to use the Central Limit theorem in part (ii) (b).

3 The times, in minutes, taken by people to complete a walk are normally distributed with mean \(\mu\). The times, \(t\) minutes, for a random sample of 80 people were summarised as follows. $$\Sigma t = 7220 \quad \Sigma t ^ { 2 } = 656060$$

1. Calculate a \(97 \%\) confidence interval for \(\mu\).
2. Explain whether it was necessary to use the Central Limit theorem in part (i).

3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below. $$n = 60 \quad \Sigma x = 3420 \quad \Sigma x ^ { 2 } = 195200$$

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.