2.05e Hypothesis test for normal mean: known variance

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.
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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.
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7 The heights, in centimetres, of adult females in Litania have mean \(\mu\) and standard deviation \(\sigma\). It is known that in 2004 the values of \(\mu\) and \(\sigma\) were 163.21 and 6.95 respectively. The government claims that the value of \(\mu\) this year is greater than it was in 2004. In order to test this claim a researcher plans to carry out a hypothesis test at the \(1 \%\) significance level. He records the heights of a random sample of 300 adult females in Litania this year and finds the value of the sample mean.

1. State the probability of a Type I error. \includegraphics[max width=\textwidth, alt={}]{ff3433b0-baab-45e3-845e-56a794739bba-12_74_1577_557_322} ........................................................................................................................................ You should assume that the value of \(\sigma\) after 2004 remains at 6.95 .
2. Given that the value of \(\mu\) this year is actually 164.91 , find the probability of a Type II error.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

2 The times taken for the pupils in Ming's year group to do their English homework have a normal distribution with standard deviation 15.7 minutes. A teacher estimates that the mean time is 42 minutes. The times taken by a random sample of 3 students from the year group were 27, 35 and 43 minutes. Carry out a hypothesis test at the \(10 \%\) significance level to determine whether the teacher's estimate for the mean should be accepted, stating the null and alternative hypotheses.

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.

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.

8 The quantity, \(X\) milligrams per litre, of silicon dioxide in a certain brand of mineral water is a random variable with distribution \(\mathrm { N } \left( \mu , 5.6 ^ { 2 } \right)\).

1. A random sample of 80 observations of \(X\) has sample mean 100.7. Test, at the \(1 \%\) significance level, the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 102\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 102\).
2. The test is redesigned so as to meet the following conditions.
   • The hypotheses are \(\mathrm { H } _ { 0 } : \mu = 102\) and \(\mathrm { H } _ { 1 } : \mu < 102\).
   • The significance level is \(1 \%\).
   • The probability of making a Type II error when \(\mu = 100\) is to be (approximately) 0.05 .
   The sample size is \(n\), and the critical region is \(\bar { X } < c\), where \(\bar { X }\) denotes the sample mean.
   1. Show that \(n\) and \(c\) satisfy (approximately) the equation \(102 - c = \frac { 13.0256 } { \sqrt { n } }\).
   2. Find another equation satisfied by \(n\) and \(c\).
   3. Hence find the values of \(n\) and \(c\).

5 Over a long period the number of visitors per week to a stately home was known to have the distribution \(\mathrm { N } \left( 500,100 ^ { 2 } \right)\). After higher car parking charges were introduced, a sample of four randomly chosen weeks gave a mean number of visitors per week of 435 . You should assume that the number of visitors per week is still normally distributed with variance \(100 ^ { 2 }\).

1. Test, at the \(10 \%\) significance level, whether there is evidence that the mean number of visitors per week has fallen.
2. Explain why it is necessary to assume that the distribution of the number of visitors per week (after the introduction of higher charges) is normal in order to carry out the test.

7 Three independent researchers, \(A , B\) and \(C\), carry out significance tests on the power consumption of a manufacturer's domestic heaters. The power consumption, \(X\) watts, is a normally distributed random variable with mean \(\mu\) and standard deviation 60. Each researcher tests the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 4000\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu > 4000\). Researcher \(A\) uses a sample of size 50 and a significance level of \(5 \%\).

1. Find the critical region for this test, giving your answer correct to 4 significant figures. In fact the value of \(\mu\) is 4020 .
2. Calculate the probability that Researcher \(A\) makes a Type II error.
3. Researcher \(B\) uses a sample bigger than 50 and a significance level of \(5 \%\). Explain whether the probability that Researcher \(B\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
4. Researcher \(C\) uses a sample of size 50 and a significance level bigger than \(5 \%\). Explain whether the probability that Researcher \(C\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
5. State with a reason whether it is necessary to use the Central Limit Theorem at any point in this question.

8 A random variable \(Y\) is normally distributed with mean \(\mu\) and variance 12.25. Two statisticians carry out significance tests of the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0\).

1. Statistician \(A\) uses the mean \(\bar { Y }\) of a sample of size 23, and the critical region for his test is \(\bar { Y } > 64.20\). Find the significance level for \(A\) 's test.
2. Statistician \(B\) uses the mean of a sample of size 50 and a significance level of \(5 \%\).
   1. Find the critical region for \(B\) 's test.
   2. Given that \(\mu = 65.0\), find the probability that \(B\) 's test results in a Type II error.
   3. Given that, when \(\mu = 65.0\), the probability that \(A\) 's test results in a Type II error is 0.1365 , state with a reason which test is better.

7 An examination board is developing a new syllabus and wants to know if the question papers are the right length. A random sample of 50 candidates was given a pre-test on a dummy paper. The times, \(t\) minutes, taken by these candidates to complete the paper can be summarised by \(n = 50\), \(\sum \boldsymbol { t } = \mathbf { 4 0 5 0 }\), \(\sum \boldsymbol { t } ^ { \mathbf { 2 } } \boldsymbol { = } \mathbf { 3 2 9 8 0 0 }\).
Assume that times are normally distributed.
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1. Estimate the proportion of candidates that could not complete the paper within 90 minutes. [6]
2. Test, at the \(10 \%\) significance level, whether the mean time for all candidates to complete this paper is \(\mathbf { 8 0 }\) minutes. Use a two-tail test. [7]
3. Explain whether the assumption that times are normally distributed is necessary in answering
   1. part (i),
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   2. part (ii). [2]

3 A company has a fleet of identical vans. Company policy is to replace all of the tyres on a van as soon as any one of them is worn out. The random variable \(X\) represents the number of miles driven before the tyres on a van are replaced. \(X\) is Normally distributed with mean 27500 and standard deviation 4000.

1. Find \(\mathrm { P } ( X > 25000 )\).
2. 10 vans in the fleet are selected at random. Find the probability that the tyres on exactly 7 of them last for more than 25000 miles.
3. The tyres of \(99 \%\) of vans last for more than \(k\) miles. Find the value of \(k\). A tyre supplier claims that a different type of tyre will have a greater mean lifetime. A random sample of 15 vans is fitted with these tyres. For each van, the number of miles driven before the tyres are replaced is recorded. A hypothesis test is carried out to investigate the claim. You may assume that these lifetimes are also Normally distributed with standard deviation 4000.
4. Write down suitable null and alternative hypotheses for the test.
5. For the 15 vans, it is found that the mean lifetime of the tyres is 28630 miles. Carry out the test at the \(5 \%\) level.

5 The time \(T\) seconds needed for a computer to be ready to use, from the moment it is switched on, is a normally distributed random variable with standard deviation 5 seconds. The specification of the computer says that the population mean time should be not more than 30 seconds.

1. A test is carried out, at the \(5 \%\) significance level, of whether the specification is being met, using the mean \(\bar { t }\) of a random sample of 10 times.
   1. Find the critical region for the test, in terms of \(\bar { t }\).
   2. Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.
   3. Because of system degradation and memory load, the population mean time \(\mu\) seconds increases with the number of months of use, \(m\). A formula for \(\mu\) in terms of \(m\) is \(\mu = 20 + 0.6 m\). Use this formula to find the value of \(m\) for which the probability that the test results in rejection of the null hypothesis is 0.5 .

3 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). A hypothesis test is carried out of \(\mathrm { H } _ { 0 } : \mu = 20.0\) against \(\mathrm { H } _ { 1 } : \mu < 20.0\), at the \(1 \%\) level of significance, based on the mean of a sample of size 16. Given that in fact \(\mu = 15.0\), find the probability that the test results in a Type II error.

8 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). A test is carried out, at the \(5 \%\) significance level, of \(\mathrm { H } _ { 0 } : \mu = 30\) against \(\mathrm { H } _ { 1 } : \mu > 30\), based on a random sample of size 18 .

1. Find the critical region for the test.
2. If \(\mu = 30\) and the outcome of the test is that \(\mathrm { H } _ { 0 }\) is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that \(\mathrm { H } _ { 0 }\) is rejected. It is known that the value of \(\mu\) remains the same throughout these 20 tests.
3. Find the probability that \(\mathrm { H } _ { 0 }\) is rejected at least 4 times if \(\mu = 30\). Hence state whether you think that \(\mu = 30\), giving a reason.
4. Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of \(\mu\), giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

3 Intensity of light is measured in lumens. The random variable \(X\) represents the intensity of the light from a standard 100 watt light bulb. \(X\) is Normally distributed with mean 1720 and standard deviation 90. You may assume that the intensities for different bulbs are independent.

1. Show that \(\mathrm { P } ( X < 1700 ) = 0.4121\).
2. These bulbs are sold in packs of 4 . Find the probability that the intensities of exactly 2 of the 4 bulbs in a randomly chosen pack are below 1700 lumens.
3. Use a suitable approximating distribution to find the probability that the intensities of at least 20 out of 40 randomly selected bulbs are below 1700 lumens. A manufacturer claims that the average intensity of its 25 watt low energy light bulbs is 1720 lumens. A consumer organisation suspects that the true figure may be lower than this. The intensities of a random sample of 20 of these bulbs are measured. A hypothesis test is then carried out to check the claim.
4. Write down a suitable null hypothesis and explain briefly why the alternative hypothesis should be \(\mathrm { H } _ { 1 } : \mu < 1720\). State the meaning of \(\mu\).
5. Given that the standard deviation of the intensity of such bulbs is 90 lumens and that the mean intensity of the sample of 20 bulbs is 1703 lumens, carry out the test at the \(5 \%\) significance level.

3 In a men's cycling time trial, the times are modelled by the random variable \(X\) minutes which is Normally distributed with mean 63 and standard deviation 5.2.

1. Find $$\begin{aligned} & \text { (A) } \mathrm { P } ( X < 65 ) \text {, } \\ & \text { (B) } \mathrm { P } ( 60 < X < 65 ) \text {. } \end{aligned}$$
2. Find the probability that 5 riders selected at random all record times between 60 and 65 minutes.
3. A competitor aims to be in the fastest \(5 \%\) of entrants (i.e. those with the lowest times). Find the maximum time that he can take. It is suggested that holding the time trial on a new course may result in lower times. To investigate this, a random sample of 15 competitors is selected. These 15 competitors do the time trial on the new course. The mean time taken by these riders is 61.7 minutes. You may assume that times are Normally distributed and the standard deviation is still 5.2 minutes. A hypothesis test is carried out to investigate whether times on the new course are lower.
4. Write down suitable null and alternative hypotheses for the test. Carry out the test at the 5\% significance level.

3 The wing lengths of native English male blackbirds, measured in mm , are Normally distributed with mean 130.5 and variance 11.84.

1. Find the probability that a randomly selected native English male blackbird has a wing length greater than 135 mm .
2. Given that \(1 \%\) of native English male blackbirds have wing length more than \(k \mathrm {~mm}\), find the value of \(k\).
3. Find the probability that a randomly selected native English male blackbird has a wing length which is 131 mm correct to the nearest millimetre. It is suspected that Scandinavian male blackbirds have, on average, longer wings than native English male blackbirds. A random sample of 20 Scandinavian male blackbirds has mean wing length 132.4 mm . You may assume that wing lengths in this population are Normally distributed with variance \(11.84 \mathrm {~mm} ^ { 2 }\).
4. Carry out an appropriate hypothesis test, at the \(5 \%\) significance level.
5. Discuss briefly one advantage and one disadvantage of using a \(10 \%\) significance level rather than a \(5 \%\) significance level in hypothesis testing in general.

6 The amount of caffeine in a randomly selected cup of coffee dispensed by a machine has a normal distribution. The amount of caffeine in each of a random sample of 25 cups was measured. The sample mean was 110.4 mg and the unbiased estimate of the population variance was \(50.42 \mathrm { mg } ^ { 2 }\). Calculate a 90\% confidence interval for the mean amount of caffeine dispensed.

8 In a crossword competition the times, \(x\) minutes, taken by a random sample of 6 entrants to complete a crossword are summarised as follows. $$\Sigma x = 210.9 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 151.2$$ The time to complete a crossword has a normal distribution with mean \(\mu\) minutes. Calculate a \(95 \%\) confidence interval for \(\mu\). Assume now that the standard deviation of the population is known to be 5.6 minutes. Find the smallest sample size that would lead to a \(95 \%\) confidence interval for \(\mu\) of width at most 5 minutes.

10 A certain forest contains only trees of a particular species. Dipak wished to take a random sample of 5 trees from the forest. He numbered the trees from 1 to 784. Then, using his calculator, he generated the random digits 14781049 . Using these digits, Dipak formed 5 three-digit numbers. He took the first, second and third digits, followed by the second, third and fourth digits and so on. In this way he obtained the following list of numbers for his sample. $$\begin{array} { l l l l l } 147 & 478 & 781 & 104 & 49 \end{array}$$

1. Explain why Dipak omitted the number 810 from his list.
2. Explain why Dipak's sample is not random. The mean height of all trees of this species is known to be 4.2 m . Dipak wishes to test whether the mean height of trees in the forest is less than 4.2 m . He now uses a correct method to choose a random sample of 50 trees and finds that their mean height is 4.0 m . It is given that the standard deviation of trees in the forest is 0.8 m .
3. Carry out the test at the \(2 \%\) significance level.

10 The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable \(X\), where \(X\) has the distribution \(\mathrm { N } ( \mu , 0.0000409 )\). In the past the value of \(\mu\) has been 0.0340 . This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the 5\% significance level, whether the mean level of pollutant has changed.

11 Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

1. State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
2. Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.

   Mean	Standard deviation	Median	Lower quartile	Upper quartile
   21.0	4.20	20.5	18.0	22.9

3. Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
4. Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the 5\% significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.

10 In the past, the time spent in minutes, by customers in a certain library had mean 32.5 and standard deviation 8.2. Following a change of layout in the library, the mean time spent in the library by a random sample of 50 customers is found to be 34.5 minutes. Assuming that the standard deviation remains at 8.2 , test at the \(5 \%\) significance level whether the mean time spent by customers in the library has changed.

1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml

Given that 15\% of bottles contain less than 24.63 ml

1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\) The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16 ml Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml
3. Test Hannah's belief at the \(5 \%\) level of significance. You should state your hypotheses clearly.