5.05c Hypothesis test: normal distribution for population mean

7 The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$ Carry out the test at the 10\% significance level.

5 The lectures in a mathematics department are scheduled to last 54 minutes, and the times of individual lectures may be assumed to have a normal distribution with mean \(\mu\) minutes and standard deviation 3.1 minutes. One of the students commented that, on average, the lectures seemed too short. To investigate this, the times for a random sample of 10 lectures were used to test the null hypothesis \(\mu = 54\) against the alternative hypothesis \(\mu < 54\) at the \(10 \%\) significance level.

1. Show that the null hypothesis is rejected in favour of the alternative hypothesis if \(\bar { x } < 52.74\), where \(\bar { x }\) minutes is the sample mean.
2. Find the probability of a Type II error given that the actual mean length of lectures is 51.5 minutes.

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.

3 A machine has produced nails over a long period of time, where the length in millimetres was distributed as \(\mathrm { N } ( 22.0,0.19 )\). It is believed that recently the mean length has changed. To test this belief a random sample of 8 nails is taken and the mean length is found to be 21.7 mm . Carry out a hypothesis test at the \(5 \%\) significance level to test whether the population mean has changed, assuming that the variance remains the same.

4 People who diet can expect to lose an average of 3 kg in a month. In a book, the authors claim that people who follow a new diet will lose an average of more than 3 kg in a month. The weight losses of the 180 people in a random sample who had followed the new diet for a month were noted. The mean was 3.3 kg and the standard deviation was 2.8 kg .

1. Test the authors' claim at the \(5 \%\) significance level, stating your null and alternative hypotheses.
2. State what is meant by a Type II error in words relating to the context of the test in part (i).

1 In Europe the diameters of women's rings have mean 18.5 mm . Researchers claim that women in Jakarta have smaller fingers than women in Europe. The researchers took a random sample of 20 women in Jakarta and measured the diameters of their rings. The mean diameter was found to be 18.1 mm . Assuming that the diameters of women's rings in Jakarta have a normal distribution with standard deviation 1.1 mm , carry out a hypothesis test at the \(2 \frac { 1 } { 2 } \%\) level to determine whether the researchers' claim is justified.

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.

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.

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 .

1 The number of new enquiries per day at an office has a Poisson distribution. In the past the mean has been 3 . Following a change of staff, the manager wishes to test, at the \(5 \%\) significance level, whether the mean has increased.

1. State the null and alternative hypotheses for this test. The manager notes the number, \(N\), of new enquiries during a certain 6 -day period. She finds that \(N = 25\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \geqslant 25 ) = 0.0683\).
2. What conclusion should she draw?

3 In a sample of 50 students at Batlin college, 18 support the football club Real Madrid.

1. Calculate an approximate \(98 \%\) confidence interval for the proportion of students at Batlin college who support Real Madrid.
2. Give one condition for this to be a reliable result.

7 The weights, \(X\) kilograms, of bags of carrots are normally distributed. The mean of \(X\) is \(\mu\). An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$

1. Carry out the test, at the \(10 \%\) significance level.
2. You may now assume that the population variance of \(X\) is 1.85 . The inspector weighs another random sample of 200 bags and carries out the same test at the \(10 \%\) significance level.
   1. State the meaning of a Type II error in this context.
   2. Given that \(\mu = 2.12\), show that the probability of a Type II error is 0.652 , correct to 3 significant figures.

5

1. Deng wishes to test whether a certain coin is biased so that it is more likely to show Heads than Tails. He throws it 12 times. If it shows Heads more than 9 times, he will conclude that the coin is biased. Calculate the significance level of the test.
2. Deng throws another coin 100 times in order to test, at the \(5 \%\) significance level, whether it is biased towards Heads. Find the rejection region for this test.

6 Last year Samir found that the time for his journey to work had mean 45.7 minutes and standard deviation 3.2 minutes. Samir wishes to test whether his journey times have increased this year. He notes the times, in minutes, for a random sample of 8 journeys this year with the following results. $$\begin{array} { l l l l l l l l } 46.2 & 41.7 & 49.2 & 47.1 & 47.2 & 48.4 & 53.7 & 45.5 \end{array}$$ It may be assumed that the population of this year's journey times is normally distributed with standard deviation 3.2 minutes.

1. State, with a reason, whether Samir should use a one-tail or a two-tail test.
2. Show that there is no evidence at the \(5 \%\) significance level that Samir's mean journey time has increased.
3. State, with a reason, which one of the errors, Type I or Type II, might have been made in carrying out the test in part (ii).

2 The times taken by students to complete a task are normally distributed with standard deviation 2.4 minutes. A lecturer claims that the mean time is 17.0 minutes. The times taken by a random sample of 5 students were 17.8, 22.4, 16.3, 23.1 and 11.4 minutes. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether the lecturer's claim should be accepted.

7 Leila suspects that a particular six-sided die is biased so that the probability, \(p\), that it will show a six is greater than \(\frac { 1 } { 6 }\). She tests the die by throwing it 5 times. If it shows a six on 3 or more throws she will conclude that it is biased.

1. State what is meant by a Type I error in this situation and calculate the probability of a Type I error.
2. Assuming that the value of \(p\) is actually \(\frac { 2 } { 3 }\), calculate the probability of a Type II error. Leila now throws the die 80 times and it shows a six on 50 throws.
3. Calculate an approximate \(96 \%\) confidence interval for \(p\).

3 The heights of a certain variety of plant have been found to be normally distributed with mean 75.2 cm and standard deviation 5.7 cm . A biologist suspects that pollution in a certain region is causing the plants to be shorter than usual. He takes a random sample of \(n\) plants of this variety from this region and finds that their mean height is 73.1 cm . He then carries out an appropriate hypothesis test.

1. He finds that the value of the test statistic \(z\) is - 1.563 , correct to 3 decimal places. Calculate the value of \(n\). State an assumption necessary for your calculation.
2. Use this value of the test statistic to carry out the hypothesis test at the 6\% significance level.

6 The number of cases of asthma per month at a clinic has a Poisson distribution. In the past the mean has been 5.3 cases per month. A new treatment is introduced. In order to test at the \(5 \%\) significance level whether the mean has decreased, the number of cases in a randomly chosen month is noted.

1. Find the critical region for the test and, given that the number of cases is 2 , carry out the test.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. At another clinic the mean number of cases of asthma per month has the independent distribution \(\mathrm { Po } ( 13.1 )\). Assuming that the mean for the first clinic is still 5.3, use a suitable approximating distribution to estimate the probability that the total number of cases in the two clinics in a particular month is more than 20.

7 In the past the weekly profit at a store had mean \(\\) 34600\( and standard deviation \)\\( 4500\). Following a change of ownership, the mean weekly profit for 90 randomly chosen weeks was \(\\) 35400$.

1. Stating a necessary assumption, test at the \(5 \%\) significance level whether the mean weekly profit has increased.
2. State, with a reason, whether it was necessary to use the Central Limit theorem in part (i). The mean weekly profit for another random sample of 90 weeks is found and the same test is carried out at the 5\% significance level.
3. State the probability of a Type I error.
4. Given that the population mean weekly profit is now \(\\) 36500$, calculate the probability of a Type II error.

3 The lengths, in centimetres, of rods produced in a factory have mean \(\mu\) and standard deviation 0.2. The value of \(\mu\) is supposed to be 250 , but a manager claims that one machine is producing rods that are too long on average. A random sample of 40 rods from this machine is taken and the sample mean length is found to be 250.06 cm . Test at the \(5 \%\) significance level whether the manager's claim is justified.

4 The weights, \(X\) kilograms, of rabbits in a certain area have population mean \(\mu \mathrm { kg }\). A random sample of 100 rabbits from this area was taken and the weights are summarised by $$\Sigma x = 165 , \quad \Sigma x ^ { 2 } = 276.25 .$$ Test at the \(5 \%\) significance level the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 1.6\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 1.6\).

4 In the past, the time taken by vehicles to drive along a particular stretch of road has had mean 12.4 minutes and standard deviation 2.1 minutes. Some new signs are installed and it is expected that the mean time will increase. In order to test whether this is the case, the mean time for a random sample of 50 vehicles is found. You may assume that the standard deviation is unchanged.

1. The mean time for the sample of 50 vehicles is found to be 12.9 minutes. Test at the \(2.5 \%\) significance level whether the population mean time has increased.
2. State what is meant by a Type II error in this context.
3. State what extra piece of information would be needed in order to find the probability of a Type II 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).

5 The mass, in kilograms, of rocks in a certain area has mean 14.2 and standard deviation 3.1.

1. Find the probability that the mean mass of a random sample of 50 of these rocks is less than 14.0 kg .
2. Explain whether it was necessary to assume that the population of the masses of these rocks is normally distributed.
3. A geologist suspects that rocks in another area have a mean mass which is less than 14.2 kg . A random sample of 100 rocks in this area has sample mean 13.5 kg . Assuming that the standard deviation for rocks in this area is also 3.1 kg , test at the \(2 \%\) significance level whether the geologist is correct.