5.05c Hypothesis test: normal distribution for population mean

5 In 2001, the mean height of students at the end of their final year at Bright Hope Secondary School was 165 centimetres. In 2010, David and James selected a random sample of 100 students who were at the end of their final year at this school. They recorded these students' heights, \(x\) centimetres, and found that \(\bar { x } = 167.1\) and \(s ^ { 2 } = 101.2\). To investigate the claim that the mean height had increased since 2001, David and James each correctly conducted a hypothesis test. They used the same null hypothesis and the same alternative hypothesis. However, David used a \(5 \%\) level of significance whilst James used a \(1 \%\) level of significance.

1. 1. Write down the null and alternative hypotheses that both David and James used.
      (l mark)
   2. Determine the outcome of each of the two hypothesis tests, giving each conclusion in context.
   3. State why both David and James made use of the Central Limit Theorem in their hypothesis tests.
2. It was later found that, in 2010, the mean height of students at the end of their final year at Bright Hope Secondary School was actually 165 centimetres. Giving a reason for your answer in each case, determine whether a Type I error or a Type II error or neither was made in the hypothesis test conducted by:
   1. David;
   2. James.

2

1. A particular bowling club has a large number of members. Their ages may be modelled by a normal random variable, \(X\), with standard deviation 7.5 years. On 30 June 2010, Ted, the club secretary, concerned about the ageing membership, selected a random sample of 16 members and calculated their mean age to be 65.0 years.
   1. Carry out a hypothesis test, at the \(5 \%\) level of significance, to determine whether the mean age of the club's members has changed from its value of 61.4 years on 30 June 2000.
   2. Comment on the likely number of members who were under the age of 25 on 30 June 2010, giving a numerical reason for your answer.
2. During 2011, in an attempt to encourage greater participation in the sport, the club ran a recruitment drive. After the recruitment drive, the ages of members of the bowling club may be modelled by a normal random variable, \(Y\) years, with mean \(\mu\) and standard deviation \(\sigma\). The ages, \(y\) years, of a random sample of 12 such members are summarised below. $$\sum y = 702 \quad \text { and } \quad \sum ( y - \bar { y } ) ^ { 2 } = 88.25$$
   1. Construct a \(90 \%\) confidence interval for \(\mu\), giving the limits to one decimal place.
   2. Use your confidence interval to state, with a reason, whether the recruitment drive lowered the average age of the club's members.

7 A factory produces 3-litre bottles of mineral water. The volume of water in a bottle has previously had a mean value of 3020 millilitres. Following a stoppage for maintenance, the volume of water, \(x\) millilitres, in each of a random sample of 100 bottles is measured and the following data obtained, where \(y = x - 3000\). $$\sum y = 1847.0 \quad \sum ( y - \bar { y } ) ^ { 2 } = 6336.00$$

1. Carry out a hypothesis test, at the \(5 \%\) significance level, to investigate whether the mean volume of water in a bottle has changed.
   (8 marks)
2. Subsequent measurements establish that the mean volume of water in a bottle produced by the factory after the stoppage is 3020 millilitres. State whether a Type I error, a Type II error or no error was made when carrying out the test in part (a).
   (l mark)

6 The contents, in millilitres, of cartons of milk produced at Kream Dairies, can be modelled by a normal distribution with mean 568 and variance \(\sigma ^ { 2 }\). After receiving several complaints from their customers who thought that the average content of the cartons had been reduced, the production manager of Kream Dairies decided to investigate. A random sample of 8 cartons of milk was taken, revealing the following contents, in millilitres. $$\begin{array} { l l l l l l l l } 560 & 568 & 561 & 562 & 564 & 567 & 565 & 563 \end{array}$$ Investigate, at the \(1 \%\) level of significance, whether the average content of cartons of milk is less than 568 millilitres.
(10 marks)

8 The mean age of people attending a large concert is claimed to be 35 years. A random sample of 100 people attending the concert was taken and their mean age was found to be 37.9 years.

1. Given that the standard deviation of the ages of the people attending the concert is 12 years, test, at the \(1 \%\) level of significance, the claim that the mean age is 35 years.
   (7 marks)
2. Explain, in the context of this question, the meaning of a Type II error.
   (2 marks)

6 The lifetime, \(X\) hours, of Everwhite camera batteries is normally distributed. The manufacturer claims that the mean lifetime of these batteries is 100 hours.

1. The members of a photography club suspect that the batteries do not last as long as is claimed by the manufacturer. In order to investigate their suspicion, the members test a random sample of five of these batteries and find the lifetimes, in hours, to be as follows: $$\begin{array} { l l l l l } 85 & 92 & 100 & 95 & 99 \end{array}$$ Test the members' suspicion at the \(5 \%\) level of significance.
2. The manufacturer, believing that the mean lifetime of these batteries has not changed from 100 hours, decides to determine the lifetime, \(x\) hours, of each of a random sample of 80 Everwhite camera batteries. The manufacturer obtains the following results, where \(\bar { x }\) denotes the sample mean: $$\sum x = 8080 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 6399$$ Test the manufacturer's belief at the \(5 \%\) level of significance.

3 Alan's company produces packets of crisps. The standard deviation of the weight of a packet of crisps is known to be 2.5 grams. Alan believes that, due to the extra demand on the production line at a busy time of the year, the mean weight of packets of crisps is not equal to the target weight of 34.5 grams. In an experiment set up to investigate Alan's belief, the weights of a random sample of 50 packets of crisps were recorded. The mean weight of this sample is 35.1 grams. Investigate Alan's belief at the \(5 \%\) level of significance.

5 The weight of fat in a digestive biscuit is known to be normally distributed.
Pat conducted an experiment in which she measured the weight of fat, \(x\) grams, in each of a random sample of 10 digestive biscuits, with the following results: $$\sum x = 31.9 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1.849$$

1. 1. Construct a \(99 \%\) confidence interval for the mean weight of fat in digestive biscuits.
   2. Comment on a claim that the mean weight of fat in digestive biscuits is 3.5 grams.
2. If 200 such \(99 \%\) confidence intervals were constructed, how many would you expect not to contain the population mean?

6 The management of the Wellfit gym claims that the mean cholesterol level of those members who have held membership of the gym for more than one year is 3.8 . A local doctor believes that the management's claim is too low and investigates by measuring the cholesterol levels of a random sample of 7 such members of the Wellfit gym, with the following results: $$\begin{array} { l l l l l l l } 4.2 & 4.3 & 3.9 & 3.8 & 3.6 & 4.8 & 4.1 \end{array}$$ Is there evidence, at the \(5 \%\) level of significance, to justify the doctor's belief that the mean cholesterol level is greater than the management's claim? State any assumption that you make.

5

1. The lifetime of a new 16-watt energy-saving light bulb may be modelled by a normal random variable with standard deviation 640 hours. A random sample of 25 bulbs, taken by the manufacturer from this distribution, has a mean lifetime of 19700 hours. Carry out a hypothesis test, at the \(1 \%\) level of significance, to determine whether the mean lifetime has changed from 20000 hours.
2. The lifetime of a new 11-watt energy-saving light bulb may be modelled by a normal random variable with mean \(\mu\) hours and standard deviation \(\sigma\) hours. The manufacturer claims that the mean lifetime of these energy-saving bulbs is 10000 hours. Christine, from a consumer organisation, believes that this is an overestimate. To investigate her belief, she carries out a hypothesis test at the \(5 \%\) level of significance based on the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 10000\).
   1. State the alternative hypothesis that should be used by Christine in this test.
   2. From the lifetimes of a random sample of 16 bulbs, Christine finds that \(s = 500\) hours. Determine the range of values for the sample mean which would lead Christine not to reject her null hypothesis.
   3. It was later revealed that \(\mu = 10000\). State which type of error, if any, was made by Christine if she concluded that her null hypothesis should not be rejected.
      (l mark)

2 The times taken to complete a round of golf at Slowpace Golf Club may be modelled by a random variable with mean \(\mu\) hours and standard deviation 1.1 hours. Julian claims that, on average, the time taken to complete a round of golf at Slowpace Golf Club is greater than 4 hours. The times of 40 randomly selected completed rounds of golf at Slowpace Golf Club result in a mean of 4.2 hours.

1. Investigate Julian's claim at the \(5 \%\) level of significance.
2. If the actual mean time taken to complete a round of golf at Slowpace Golf Club is 4.5 hours, determine whether a Type I error, a Type II error or neither was made in the test conducted in part (a). Give a reason for your answer.

6 A supermarket buys pears from a local supplier. The supermarket requires the mean weight of the pears to be at least 175 grams. William, the fresh-produce manager at the supermarket, suspects that the latest batch of pears delivered does not meet this requirement.

1. William weighs a random sample of 6 pears, obtaining the following weights, in grams. $$\begin{array} { l l l l l l } 160.6 & 155.4 & 181.3 & 176.2 & 162.3 & 172.8 \end{array}$$ Previous batches of pears have had weights that could be modelled by a normal distribution with standard deviation 9.4 grams. Assuming that this still applies, show that a hypothesis test at the \(5 \%\) level of significance supports William's suspicion.
   (7 marks)
2. William then weighs a random sample of 20 pears. The mean of this sample is 169.4 grams and \(s = 11.2\) grams, where \(s ^ { 2 }\) is an unbiased estimate of the population variance. Assuming that the population from which this sample is taken has a normal distribution but with unknown standard deviation, test William's suspicion at the \(\mathbf { 1 \% }\) level of significance.
3. Give a reason why the probability of a Type I error occurring was smaller when conducting the test in part (b) than when conducting the test in part (a).

6 South Riding Alarms (SRA) maintains household burglar-alarm systems. The company aims to carry out an annual service of a system in a mean time of 20 minutes.
Technicians who carry out an annual service must record the times at which they start and finish the service.

1. Gary is employed as a technician by SRA and his manager, Rajul, calculates the times taken for 8 annual services carried out by Gary. The results, in minutes, are as follows. $$\begin{array} { l l l l l l l l } 24 & 25 & 29 & 16 & 18 & 27 & 19 & 23 \end{array}$$ Assume that these times may be regarded as a random sample from a normal distribution. Carry out a hypothesis test, at the \(10 \%\) significance level, to examine whether the mean time for an annual service carried out by Gary is 20 minutes.
   [0pt] [8 marks]
2. Rajul suspects that Gary may be taking longer than 20 minutes on average to carry out an annual service. Rajul therefore calculates the times taken for 100 annual services carried out by Gary. Assume that these times may also be regarded as a random sample from a normal distribution but with a standard deviation of 4.6 minutes. Find the highest value of the sample mean which would not support Rajul's suspicion at the \(5 \%\) significance level. Give your answer to two decimal places.
   [0pt] [4 marks] \(7 \quad\) A continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 4 } { 5 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 20 } ( x - 3 ) ( 3 x - 11 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
   1. Find \(\mathrm { P } ( X < 1 )\).
   2. 1. Show that, for \(1 \leqslant x \leqslant 3\), the cumulative distribution function, \(\mathrm { F } ( x )\), is given by $$\mathrm { F } ( x ) = \frac { 1 } { 20 } \left( x ^ { 3 } - 10 x ^ { 2 } + 33 x - 16 \right)$$
      2. Hence verify that the median value of \(X\) lies between 1.13 and 1.14 .
         [0pt] [3 marks] QUESTION
         PART Answer space for question 7
         REFERENCE REFERENCE
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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]

3. A die is rolled 60 times, and results in 16 sixes.

1. Use a suitable approximation to test, at the \(5 \%\) significance level, whether the probability of scoring a six is \(\frac { 1 } { 6 }\) or not. State your hypotheses clearly.
2. Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than \(\frac { 1 } { 6 }\). Carry out this modified test.

6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.

1. Find the significance level of this test.
2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
   (6 marks)

3 Pitted black olives in brine are sold in jars labelled " 340 grams net weight". Two machines, A and B, independently fill these jars with olives before the brine is added. The weight, \(X\) grams, of olives delivered by machine A may be modelled by a normal distribution with mean \(\mu _ { X }\) and standard deviation 4.5. The weight, \(Y\) grams, of olives delivered by machine B may be modelled by a normal distribution with mean \(\mu _ { Y }\) and standard deviation 5.7. The mean weight of olives from a random sample of 10 jars filled by machine A is found to be 157 grams, whereas that from a random sample of 15 jars filled by machine \(B\) is found to be 162 grams. Test, at the \(1 \%\) level of significance, the hypothesis that \(\mu _ { X } = \mu _ { Y }\).
(6 marks)

4 Holly, a horticultural researcher, believes that the mean height of stems on Tahiti daffodils exceeds that on Jetfire daffodils by more than 15 cm . She measures the heights, \(x\) centimetres, of stems on a random sample of 65 Tahiti daffodils and finds that their mean, \(\bar { x }\), is 40.7 and that their standard deviation, \(s _ { x }\), is 3.4 . She also measures the heights, \(y\) centimetres, of stems on a random sample of 75 Jetfire daffodils and finds that their mean, \(\bar { y }\), is 24.4 and that their standard deviation, \(s _ { y }\), is 2.8 . Investigate, at the \(1 \%\) level of significance, Holly's belief.

2 Rodney and Derrick, two independent fruit and vegetable market stallholders, sell punnets of locally-grown raspberries from their stalls during June and July. The following information, based on independent random samples, was collected as part of an investigation by Trading Standards Officers.

1. Construct a \(99 \%\) confidence interval for the difference between the mean weight of raspberries in a punnet sold by Rodney and the mean weight of raspberries in a punnet sold by Derrick.
2. What can be concluded from your confidence interval?
3. In addition to weight, state one other factor that may influence whether customers buy raspberries from Rodney or from Derrick.
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6

1. A district council claimed that more than 80 per cent of the complaints that it received about the delivery of its services were answered to the satisfaction of complainants before reaching formal status. An analysis of a random sample of 175 complaints revealed that 28 reached formal status.
   1. Construct an approximate \(95 \%\) confidence interval for the proportion of complaints that reach formal status.
   2. Hence comment on the council's claim.
2. The district council also claimed that less than 40 per cent of all formal complaints were due to a failing in the delivery of its services. An analysis of the 50 formal complaints received during 2007/08 showed that 16 were due to a failing in the delivery of its services.
   1. Using an exact test, investigate the council's claim at the \(10 \%\) level of significance. The 50 formal complaints received during 2007/08 may be assumed to be a random sample.
   2. Determine the critical value for your test in part (b)(i).
   3. In fact, only 25 per cent of all formal complaints were due to a failing in the delivery of the council's services. Determine the probability of a Type II error for a test of the council's claim at the \(10 \%\) level of significance and based on the analysis of a random sample of 50 formal complaints.
      (4 marks)
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8 The tensile strength of rope is measured in kilograms. The standard deviation of the tensile strength of a particular design of 10 mm diameter rope is known to be 285 kilograms. A retail organisation, which buys such rope from two manufacturers, A and B , wishes to compare their ropes for mean tensile strength. The mean tensile strength, \(\bar { x }\), of a random sample of 80 lengths from manufacturer A was 3770 kilograms. The mean tensile strength, \(\bar { y }\), of a random sample of 120 lengths from manufacturer B was 3695 kilograms.

1. 1. Test, at the \(5 \%\) level of significance, the hypothesis that there is no difference between the mean tensile strength of rope from manufacturer A and that of rope from manufacturer B.
   2. Why was it not necessary to know the distributions of tensile strength in order for your test in part (a)(i) to be valid?
2. 1. Deduce that, for your test in part (a)(i), the critical values of \(( \bar { x } - \bar { y } )\) are \(\pm 80.63\), correct to two decimal places.
   2. In fact, the mean tensile strength of rope from manufacturer A exceeds that of rope from manufacturer B by 125 kilograms. Determine the probability of a Type II error for a test of the hypothesis in part (a)(i) at the \(5 \%\) level of significance, based upon a random sample of 80 lengths from manufacturer A and a random sample of 120 lengths from manufacturer B. (4 marks)

2 As part of a comparison of two varieties of cucumber, Fanfare and Marketmore, random samples of harvested cucumbers of each variety were selected and their lengths measured, in centimetres. The results are summarised in the table.

1. Test, at the \(1 \%\) level of significance, the hypothesis that there is no difference between the mean length of harvested Fanfare cucumbers and that of harvested Marketmore cucumbers.
2. In addition to length, name one other characteristic of cucumbers that could be used for comparative purposes.

4 An analysis of a sample of 250 patients visiting a medical centre showed that 38 per cent were aged over 65 years. An analysis of a sample of 100 patients visiting a dental practice showed that 21 per cent were aged over 65 years. Assume that each of these two samples has been randomly selected.
Investigate, at the \(5 \%\) level of significance, the hypothesis that the percentage of patients visiting the medical centre, who are aged over 65 years, exceeds that of patients visiting the dental practice, who are aged over 65 years, by more than 10 per cent.

7 It is claimed that the proportion, \(P\), of people who prefer cooked fresh garden peas to cooked frozen garden peas is greater than 0.50 .

1. In an attempt to investigate this claim, a sample of 50 people were each given an unlabelled portion of cooked fresh garden peas and an unlabelled portion of cooked frozen garden peas to taste. After tasting each portion, the people were each asked to state which of the two portions they preferred. Of the 50 people sampled, 29 preferred the cooked fresh garden peas. Assuming that the 50 people may be considered to constitute a random sample, use a binomial distribution and the \(10 \%\) level of significance to investigate the claim.
   (6 marks)
2. It was then decided to repeat the tasting in part (a) but to involve a sample of 500 , rather than 50, people. Of the 500 people sampled, 271 preferred the cooked fresh garden peas.
   1. Assuming that the 500 people may be considered to constitute a random sample, use an approximation to the distribution of the sample proportion, \(\widehat { P }\), and the \(10 \%\) level of significance to again investigate the claim.
   2. The critical value of \(\widehat { P }\) for the test in part (b)(i) is 0.529 , correct to three significant figures. It is also given that, in fact, 55 per cent of people prefer cooked fresh garden peas. Estimate the power for a test of the claim that \(P > 0.50\) based on a random sample of 500 people and using the \(10 \%\) level of significance.
      (5 marks)

2 Each household within a district council's area has two types of wheelie-bin: a black one for general refuse and a green one for garden refuse. Each type of bin is emptied by the council fortnightly. The weight, in kilograms, of refuse emptied from a black bin can be modelled by the random variable \(B \sim \mathrm {~N} \left( \mu _ { B } , 0.5625 \right)\). The weight, in kilograms, of refuse emptied from a green bin can be modelled by the random variable \(G \sim \mathrm {~N} \left( \mu _ { G } , 0.9025 \right)\). The mean weight of refuse emptied from a random sample of 20 black bins was 21.35 kg . The mean weight of refuse emptied from an independent random sample of 15 green bins was 21.90 kg . Test, at the \(5 \%\) level of significance, the hypothesis that \(\mu _ { B } = \mu _ { G }\).
[0pt] [6 marks]