One-tail z-test (lower tail)

3 Batteries of type \(A\) are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type \(B\), has a shorter mean life than type \(A\) batteries.

1. Give a reason for using a sample rather than the whole population in carrying out this test.
   A random sample of 120 type \(B\) batteries are tested and it is found that their mean life is 147 hours, and an unbiased estimate of the population variance is 225 hours \(^ { 2 }\).
2. Test, at the \(2 \%\) significance level, whether type \(B\) batteries have a shorter mean life than type \(A\) batteries.
3. Calculate a \(94 \%\) confidence interval for the population mean life of type \(B\) batteries.

6 The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table.

1. Calculate an unbiased estimate for the population mean of the number of green sweets in a packet of Frutos, and show that an unbiased estimate of the population variance is 0.783 correct to 3 significant figures.
   The manufacturers of Frutos claim that the mean number of green sweets in a packet is 1.65 .
   Anji believes that the true value of the mean, \(\mu\), is less than 1.65 . She uses the results from the 200 randomly chosen packets to test the manufacturers' claim.
2. State suitable null and alternative hypotheses for the test. \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-08_2714_37_143_2008}
3. Show that the result of Anji's test is significant at the \(5 \%\) level but not at the \(1 \%\) level.
4. It is given that Anji made a Type I error. Explain how this shows that the significance level that Anji used in her test was not \(1 \%\).

3 In the past, the mean time taken by Freda for a particular daily journey was 39.2 minutes. Following the introduction of a one-way system, Freda wishes to test whether the mean time for the journey has decreased. She notes the times, \(t\) minutes, for 40 randomly chosen journeys and summarises the results as follows. $$n = 40 \quad \Sigma t = 1504 \quad \Sigma t ^ { 2 } = 57760$$

1. Calculate unbiased estimates of the population mean and variance of the new journey time.
2. Test, at the \(5 \%\) significance level, whether the population mean time has decreased.

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.

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).

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 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.

4 The mean mass of packets of sugar is supposed to be 505 g . A random sample of 10 packets filled by a certain machine was taken and the masses, in grams, were found to be as follows. $$\begin{array} { l l l l l l l l l l } 500 & 499 & 496 & 495 & 498 & 490 & 492 & 501 & 494 & 494 \end{array}$$

1. Find unbiased estimates of the population mean and variance.
   The mean mass of packets produced by this machine was found to be less than 505 g , so the machine was adjusted. Following the adjustment, the masses of a random sample of 150 packets from the machine were measured and the total mass was found to be 75660 g .
2. Given that the population standard deviation is 3.6 g , test at the \(2 \%\) significance level whether the machine is still producing packets with mean mass less than 505 g .
3. Explain why the use of the normal distribution is justified in carrying out the test in part (ii). [1]

3 It is claimed that, on average, a particular train journey takes less than 1.9 hours. The times, \(t\) hours, taken for this journey on a random sample of 50 days were recorded. The results are summarised below. $$n = 50 \quad \Sigma t = 92.5 \quad \Sigma t ^ { 2 } = 175.25$$

1. Calculate unbiased estimates of the population mean and variance.
2. Test the claim at the \(5 \%\) significance level.

3 The masses, \(m \mathrm {~kg}\), of packets of flour are normally distributed. The mean mass is supposed to be 1.01 kg . A quality control officer measures the masses of a random sample of 100 packets. The results are summarised below. $$n = 100 \quad \Sigma m = 98.2 \quad \Sigma m ^ { 2 } = 104.52$$

1. Test at the \(5 \%\) significance level whether the population mean mass is less than 1.01 kg .
2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).

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.

5 The mean solubility rating of widgets inserted into beer cans is thought to be 84.0, in appropriate units. A random sample of 50 widgets is taken. The solubility ratings, \(x\), are summarised by $$n = 50 , \quad \Sigma x = 4070 , \quad \Sigma x ^ { 2 } = 336100$$ Test, at the \(5 \%\) significance level, whether the mean solubility rating is less than 84.0 .

4 The greatest weight \(W N\) that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80 . A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N .

1. Test at the \(1 \%\) significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support.
2. State an assumption needed in carrying out the test in part (a).
3. Explain whether it is necessary to use the central limit theorem in carrying out the test.

7 Sweet pea plants grown using a standard plant food have a mean height of 1.6 m . A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$\begin{aligned} n & = 49 \\ \Sigma x & = 74.48 \\ \Sigma x ^ { 2 } & = 120.8896 \end{aligned}$$ Test, at the \(5 \%\) significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m .

1. A machine makes screws with a mean length of 30 mm and a standard deviation of 2.5 mm .

A manager claims that, following some repairs, the machine is now making screws with a mean length of less than 30 mm . The manager takes a random sample of 80 screws and finds that they have a mean length of 29.5 mm . Use a suitable test, at the \(5 \%\) level of significance, to determine whether there is evidence to support the manager's claim. State your hypotheses clearly.

1. The time, in minutes, it takes Robert to complete the puzzle in his morning newspaper each day is normally distributed with mean 18 and standard deviation 3. After taking a holiday, Robert records the times taken to complete a random sample of 15 puzzles and he finds that the mean time is 16.5 minutes. You may assume that the holiday has not changed the standard deviation of times taken to complete the puzzle.

Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there has been a reduction in the mean time Robert takes to complete the puzzle.

7. A machine fills packets with \(X\) grams of powder where \(X\) is normally distributed with mean \(\mu\). Each packet is supposed to contain 1 kg of powder. To comply with regulations, the weight of powder in a randomly selected packet should be such that \(\mathrm { P } ( X < \mu - 30 ) = 0.0005\)

1. Show that this requires the standard deviation to be 9.117 g to 3 decimal places. A random sample of 10 packets is selected from the machine. The weight, in grams, of powder in each packet is as follows 999.8991 .61000 .31006 .11008 .2997 .0993 .21000 .0997 .11002 .1
2. Assuming that the standard deviation of the population is 9.117 g , test, at the \(1 \%\) significance level, whether or not the machine is delivering packets with mean weight of less than 1 kg . State your hypotheses clearly.

6 In previous years, the marks obtained in a French test by students attending Topnotch College have been modelled satisfactorily by a normal distribution with a mean of 65 and a standard deviation of 9 . Teachers in the French department at Topnotch College suspect that this year their students are, on average, underachieving. In order to investigate this suspicion, the teachers selected a random sample of 35 students to take the French test and found that their mean score was 61.5.

1. Investigate, at the \(5 \%\) level of significance, the teachers' suspicion.
2. Explain, in the context of this question, the meaning of a Type I error.

4 Wellgrove village has a main road running through it that has a 40 mph speed limit. The villagers were concerned that many vehicles travelled too fast through the village, and so they set up a device for measuring the speed of vehicles on this main road. This device indicated that the mean speed of vehicles travelling through Wellgrove was 44.1 mph . In an attempt to reduce the mean speed of vehicles travelling through Wellgrove, life-size photographs of a police officer were erected next to the road on the approaches to the village. The speed, \(X \mathrm { mph }\), of a sample of 100 vehicles was then measured and the following data obtained. $$\sum x = 4327.0 \quad \sum ( x - \bar { x } ) ^ { 2 } = 925.71$$

1. State an assumption that must be made about the sample in order to carry out a hypothesis test to investigate whether the desired reduction in mean speed had occurred.
2. Given that the assumption that you stated in part (a) is valid, carry out such a test, using the \(1 \%\) level of significance.
3. Explain, in the context of this question, the meaning of:
   1. a Type I error;
   2. a Type II error.
      [0pt] [2 marks]

1. An engineering firm buys steel rods. The steel rods from its present supplier are known to have a mean tensile strength of \(230 \mathrm {~N} / \mathrm { mm } ^ { 2 }\).

A new supplier of steel rods offers to supply rods at a cheaper price than the present supplier. A random sample of ten rods from this new supplier gave tensile strengths, \(x \mathrm { N } / \mathrm { mm } ^ { 2 }\), which are summarised below.

1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not the rods from the new supplier have a tensile strength lower that the present supplier. (You may assume that the tensile strength is normally distributed).
2. In the light of your conclusion to part (a) write down what you would recommend the engineering firm to do.

1. A company manufactures bolts with a mean diameter of 5 mm . The company wishes to check that the diameter of the bolts has not decreased. A random sample of 10 bolts is taken and the diameters, \(x \mathrm {~mm}\), of the bolts are measured. The results are summarised below.

$$\sum x = 49.1 \quad \sum x ^ { 2 } = 241.2$$ Using a \(1 \%\) level of significance, test whether or not the mean diameter of the bolts is less than 5 mm .
(You may assume that the diameter of the bolts follows a normal distribution.)

3 David is the professional coach at the golf club where Becki is a member. He claims that, after having a series of lessons with him, the mean number of putts that Becki takes per round of golf will reduce from her present mean of 36 . After having the series of lessons with David, Becki decides to investigate his claim.
She therefore records, for each of a random sample of 50 rounds of golf, the number of putts, \(x\), that she takes to complete the round. Her results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 1730 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 784$$ Using a \(z\)-test and the \(1 \%\) level of significance, investigate David's claim.

In the past, the mean length of a particular variety of worm has been 10.3 cm, with standard deviation 2.6 cm. Following a change in the climate, it is thought that the mean length of this variety of worm has decreased. The lengths of a random sample of 100 worms of this variety are found and the mean of this sample is found to be 9.8 cm. Assuming that the standard deviation remains at 2.6 cm, carry out a test at the 2% significance level of whether the mean length has decreased. [5]

From previous years' observations, the lengths of salmon in a river were found to be normally distributed with mean 65 cm. A researcher suspects that pollution in water is restricting growth. To test this theory, she measures the length \(x\) cm of a random sample of \(n\) salmon and calculates that \(\bar{x} = 64.3\) and \(s = 4.9\), where \(s^2\) is the unbiased estimate of the population variance. She then carries out an appropriate hypothesis test.

1. Her test statistic \(z\) has a value of \(-1.807\) correct to 3 decimal places. Calculate the value of \(n\). [3]
2. Using this test statistic, carry out the hypothesis test at the 5% level of significance and state what her conclusion should be. [4]

80 randomly chosen people are asked to estimate a time interval of 60 seconds without using a watch or clock. The mean of the 80 estimates is 58.9 seconds. Previous evidence shows that the population standard deviation of such estimates is 5.0 seconds. Test, at the 5% significance level, whether there is evidence that people tend to underestimate the time interval. [7]