5.05c Hypothesis test: normal distribution for population mean

A random sample of repair times, in hours, was taken for an electronic component. The 4 observed times are shown below.
1.3
1.7
1.4
1.8

1. Calculate unbiased estimates of the mean and the variance of the population of repair times for this electronic component. The population standard deviation of the repair times for this electronic component is known to be 0.5 hours. An estimate of the population mean is required to be within 0.1 hours of its true value with a probability of at least 0.99
2. Find the minimum sample size required.
   

3 When an alarm is raised at a market town's fire station, the fire engine cannot leave until at least five fire-fighters arrive at the station. The call-out time, \(X\) minutes, is the time between an alarm being raised and the fire engine leaving the station. The value of \(X\) was recorded on a random sample of 50 occasions. The results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 286.5 \quad \sum ( x - \bar { x } ) ^ { 2 } = 45.16$$

1. Find values for the mean and standard deviation of this sample of 50 call-out times.
2. Hence construct a \(99 \%\) confidence interval for the mean call-out time.
3. The fire and rescue service claims that the station's mean call-out time is less than 5 minutes, whereas a parish councillor suggests that it is more than \(6 \frac { 1 } { 2 }\) minutes. Comment on each of these claims.

3 The height, in metres, of adult male African elephants may be assumed to be normally distributed with mean \(\mu\) and standard deviation 0.20 . The heights of a sample of 12 such elephants were measured with the following results, in metres. $$\begin{array} { l l l l l l l l l l l l } 3.37 & 3.45 & 2.93 & 3.42 & 3.49 & 3.67 & 2.96 & 3.57 & 3.36 & 2.89 & 3.22 & 2.91 \end{array}$$

1. Stating a necessary assumption, construct a \(98 \%\) confidence interval for \(\mu\). (6 marks)
2. The mean height of adult male Asian elephants is known to be 2.90 metres. Using your confidence interval, state, with a reason, what can be concluded about the mean heights of adult males in these two types of elephant.

5 The times taken by new recruits to complete an assault course may be modelled by a normal distribution with a standard deviation of 8 minutes. A group of 30 new recruits takes a total time of 1620 minutes to complete the course.

1. Calculate the mean time taken by these 30 new recruits.
2. Assuming that the 30 recruits may be considered to be a random sample, construct a \(98 \%\) confidence interval for the mean time taken by new recruits to complete the course.
3. Construct an interval within which approximately \(98 \%\) of the times taken by individual new recruits to complete the course will lie.
4. State where, if at all, in this question you made use of the Central Limit Theorem.

3 The volume, \(X\) litres, of orange juice in a 1-litre carton may be modelled by a normal distribution with unknown mean \(\mu\). The volumes, \(x\) litres, recorded to the nearest 0.01 litre, in a random sample of 100 cartons are shown in the table.

1. For the group ' \(0.98 - 1.00\) ':
   1. show that it has a mid-point of 0.99 litres;
   2. state the minimum and the maximum values of \(x\) that could be included in this group.
2. Calculate, to three decimal places, estimates of the mean and the standard deviation of these 100 volumes.
3. 1. Construct an approximate \(99 \%\) confidence interval for \(\mu\).
   2. State why use of the Central Limit Theorem was not required when calculating this confidence interval.
   3. Give a reason why the confidence interval is approximate rather than exact.
4. Give a reason in support of the claim that:
   1. \(\mu > 1\);
   2. \(\mathrm { P } ( 0.94 < X < 1.16 )\) is approximately 1 .
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7 A random sample of 50 full-time university employees was selected as part of a higher education salary survey. The annual salary in thousands of pounds, \(x\), of each employee was recorded, with the following summarised results. $$\sum x = 2290.0 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 28225.50$$ Also recorded was the fact that 6 of the 50 salaries exceeded \(\pounds 60000\).

1. 1. Calculate values for \(\bar { x }\) and \(s\), where \(s ^ { 2 }\) denotes the unbiased estimate of \(\sigma ^ { 2 }\).
   2. Hence show why the annual salary, \(X\), of a full-time university employee is unlikely to be normally distributed. Give numerical support for your answer.
2. 1. Indicate why the mean annual salary, \(\bar { X }\), of a random sample of 50 full-time university employees may be assumed to be normally distributed.
   2. Hence construct a \(99 \%\) confidence interval for the mean annual salary of full-time university employees.
3. It is claimed that the annual salaries of full-time university employees have an average which exceeds \(\pounds 55000\) and that more than \(25 \%\) of such salaries exceed £60000. Comment on each of these two claims.

6

1. The length of one-metre galvanised-steel straps used in house building may be modelled by a normal distribution with a mean of 1005 mm and a standard deviation of 15 mm . The straps are supplied to house builders in packs of 12, and the straps in a pack may be assumed to be a random sample. Determine the probability that the mean length of straps in a pack is less than one metre.
2. Tania, a purchasing officer for a nationwide house builder, measures the thickness, \(x\) millimetres, of each of a random sample of 24 galvanised-steel straps supplied by a manufacturer. She then calculates correctly that the value of \(\bar { x }\) is 4.65 mm .
   1. Assuming that the thickness, \(X \mathrm {~mm}\), of such a strap may be modelled by the distribution \(\mathrm { N } \left( \mu , 0.15 ^ { 2 } \right)\), construct a \(99 \%\) confidence interval for \(\mu\).
   2. Hence comment on the manufacturer's specification that the mean thickness of such straps is greater than 4.5 mm .

7 A machine, which cuts bread dough for loaves, can be adjusted to cut dough to any specified set weight. For any set weight, \(\mu\) grams, the actual weights of cut dough are known to be approximately normally distributed with a mean of \(\mu\) grams and a fixed standard deviation of \(\sigma\) grams. It is also known that the machine cuts dough to within 10 grams of any set weight.

1. Estimate, with justification, a value for \(\sigma\).
2. The machine is set to cut dough to a weight of 415 grams. As a training exercise, Sunita, the quality control manager, asked Dev, a recently employed trainee, to record the weight of each of a random sample of 15 such pieces of dough selected from the machine's output. She then asked him to calculate the mean and the standard deviation of his 15 recorded weights. Dev subsequently reported to Sunita that, for his sample, the mean was 391 grams and the standard deviation was 95.5 grams. Advise Sunita on whether or not each of Dev's values is likely to be correct. Give numerical support for your answers.
3. Maria, an experienced quality control officer, recorded the weight, \(y\) grams, of each of a random sample of 10 pieces of dough selected from the machine's output when it was set to cut dough to a weight of 820 grams. Her summarised results were as follows. $$\sum y = 8210.0 \quad \text { and } \quad \sum ( y - \bar { y } ) ^ { 2 } = 110.00$$ Explain, with numerical justifications, why both of these values are likely to be correct.

3

1. A sample of 50 washed baking potatoes was selected at random from a large batch.
   The weights of the 50 potatoes were found to have a mean of 234 grams and a standard deviation of 25.1 grams. Construct a \(95 \%\) confidence interval for the mean weight of potatoes in the batch.
   (4 marks)
2. The batch of potatoes is purchased by a market stallholder. He sells them to his customers by allowing them to choose any 5 potatoes for \(\pounds 1\). Give a reason why such chosen potatoes are unlikely to represent a random sample from the batch.

7 Vernon, a service engineer, is expected to carry out a boiler service in one hour.
One hour is subtracted from each of his actual times, and the resulting differences, \(x\) minutes, for a random sample of 100 boiler services are summarised in the table.

1. 1. Calculate estimates of the mean and the standard deviation of these differences.
      (4 marks)
   2. Hence deduce, in minutes, estimates of the mean and the standard deviation of Vernon's actual service times for this sample.
2. 1. Construct an approximate \(98 \%\) confidence interval for the mean time taken by Vernon to carry out a boiler service.
   2. Give a reason why this confidence interval is approximate rather than exact.
3. Vernon claims that, more often than not, a boiler service takes more than an hour and that, on average, a boiler service takes much longer than an hour. Comment, with a justification, on each of these claims.

4 Rice that can be cooked in microwave ovens is sold in packets which the manufacturer claims contain a mean weight of more than 250 grams of rice. The weight of rice in a packet may be modelled by a normal distribution. A consumer organisation's researcher weighed the contents, \(x\) grams, of each of a random sample of 50 packets. Her summarised results are: $$\bar { x } = 251.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 184.5$$

1. Show that, correct to two decimal places, \(s = 1.94\), where \(s ^ { 2 }\) denotes the unbiased estimate of the population variance.
2. 1. Construct a \(96 \%\) confidence interval for the mean weight of rice in a packet, giving the limits to one decimal place.
   2. Hence comment on the manufacturer's claim.
3. The statement '250 grams' is printed on each packet. Explain, with reference to the values of \(\bar { x }\) and \(s\), why the consumer organisation may consider this statement to be dubious.

7 The volume of bleach in a 5-litre bottle may be modelled by a random variable with a standard deviation of 75 millilitres. The volume, in litres, of bleach in each of a random sample of 36 such bottles was measured. The 36 measurements resulted in a total volume of 181.80 litres and exactly 8 bottles contained less than 5 litres.

1. Construct a 98\% confidence interval for the mean volume of bleach in a 5-litre bottle.
2. It is claimed that the mean volume of bleach in a 5-litre bottle exceeds 5 litres and also that fewer than 10 per cent of such bottles contain less than 5 litres. Comment, with numerical justification, on each of these two claims.
3. State, with justification, whether you made use of the Central Limit Theorem in constructing the confidence interval in part (a).

6 The weight, \(X\) kilograms, of sand in a bag can be modelled by a normal random variable with unknown mean \(\mu\) and known standard deviation 0.4 .

1. The sand in each of a random sample of 25 bags from a large batch is weighed. The total weight of sand in these 25 bags is found to be 497.5 kg .
   1. Construct a 98\% confidence interval for the mean weight of sand in bags in the batch.
   2. Hence comment on the claim that bags in the batch contain an average of 20 kg of sand.
   3. State why use of the Central Limit Theorem is not required in answering part (a)(i).
2. The weight, \(Y\) kilograms, of cement in a bag can be modelled by a normal random variable with mean 25.25 and standard deviation 0.35. A firm purchases 10 such bags. These bags may be considered to be a random sample.
   1. Determine the probability that the mean weight of cement in the 10 bags is less than 25 kg .
   2. Calculate the probability that the weight of cement in each of the 10 bags is more than 25 kg .
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7 For the year 2014, the table below summarises the weights, \(x\) kilograms, of a random sample of 160 women residing in a particular city who are aged between 18 years and 25 years.

1. Calculate estimates of the mean and the standard deviation of these 160 weights.
2. 1. Construct a 98\% confidence interval for the mean weight of women residing in the city who are aged between 18 years and 25 years.
   2. Hence comment on a claim that the mean weight of women residing in the city who are aged between 18 years and 25 years has increased from that of 61.7 kg in 1965.
      [0pt] [2 marks]
      
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7 The volume of water, \(V\), used by a guest in an en suite shower room at a small guest house may be modelled by a random variable with mean \(\mu\) litres and standard deviation 65 litres. A random sample of 80 guests using this shower room showed a mean usage of 118 litres of water.

1. 1. Give a numerical justification as to why \(V\) is unlikely to be normally distributed.
   2. Explain why \(\bar { V }\), the mean of a random sample of 80 observations of \(V\), may be assumed to be approximately normally distributed.
2. 1. Construct a \(98 \%\) confidence interval for \(\mu\).
   2. Hence comment on a claim that \(\mu\) is 140 .
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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.

8 Bottles of sherry nominally contain 1000 millilitres. After the introduction of a new method of filling the bottles, there is a suspicion that the mean volume of sherry in a bottle has changed. In order to investigate this suspicion, a random sample of 12 bottles of sherry is taken and the volume of sherry in each bottle is measured. The volumes, in millilitres, of sherry in these bottles are found to be Assuming that the volume of sherry in a bottle is normally distributed, investigate, at the \(5 \%\) level of significance, whether the mean volume of sherry in a bottle differs from 1000 millilitres.

3 The handicap committee of a golf club has indicated that the mean score achieved by the club's members in the past was 85.9 . A group of members believes that recent changes to the golf course have led to a change in the mean score achieved by the club's members and decides to investigate this belief. A random sample of the scores, \(x\), of 100 club members was taken and is summarised by $$\sum x = 8350 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 15321$$ where \(\bar { x }\) denotes the sample mean.
Test, at the \(5 \%\) level of significance, the group's belief that the mean score of 85.9 has changed.

5 Jasmine's French teacher states that a homework assignment should take, on average, 30 minutes to complete. Jasmine believes that he is understating the mean time that the assignment takes to complete and so decides to investigate. She records the times, in minutes, that it takes for a random sample of 10 students to complete the French assignment, with the following results: $$\begin{array} { l l l l l l l l l l } 29 & 33 & 36 & 42 & 30 & 28 & 31 & 34 & 37 & 35 \end{array}$$

1. Test, at the \(1 \%\) level of significance, Jasmine's belief that her French teacher has understated the mean time that it should take to complete the homework assignment.
2. State an assumption that you must make in order for the test used in part (a) to be valid.

1 David claims that customers have to queue at a supermarket checkout for more than 5 minutes, on average. The queuing times, \(x\) minutes, of 40 randomly selected customers result in \(\bar { x } = 5.5\) and \(s ^ { 2 } = 1.31\). Investigate, at the \(1 \%\) level of significance, David's claim.

4 A speed camera was used to measure the speed, \(V\) mph, of John's serves during a tennis singles championship. For 10 randomly selected serves, $$\sum v = 1179 \quad \text { and } \quad \sum ( v - \bar { v } ) ^ { 2 } = 1014.9$$ where \(\bar { v }\) is the sample mean.

1. Construct a \(99 \%\) confidence interval for the mean speed of John's serves at this tennis championship, stating any assumption that you make.
   (7 marks)
2. Hence comment on John's claim that, at this championship, he consistently served at speeds in excess of 130 mph .
   (1 mark)

1 Roger claims that, on average, his journey time from home to work each day is greater than 45 minutes. The times, \(x\) minutes, of 30 randomly selected journeys result in \(\bar { x } = 45.8\) and \(s ^ { 2 } = 4.8\).
Investigate Roger's claim at the \(1 \%\) level of significance.

3 Lorraine bought a new golf club. She then practised with this club by using it to hit golf balls on a golf range. After several such practice sessions, she believed that there had been no change from 190 metres in the mean distance that she had achieved when using her old club. To investigate this belief, she measured, at her next practice session, the distance, \(x\) metres, of each of a random sample of 10 shots with her new club. Her results gave $$\sum x = 1840 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1240$$ Investigate Lorraine's belief at the \(2 \%\) level of significance, stating any assumption that you make.
(7 marks)

7 Jim , a mathematics teacher, knows that the marks, \(X\), achieved by his students can be modelled by a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\). Jim selects 12 students at random and from their marks he calculates that \(\bar { x } = 64.8\) and \(s ^ { 2 } = 93.0\).

1. 1. An estimate for the standard error of the sample mean is \(d\). Show that \(d ^ { 2 } = 7.75\).
   2. Construct an \(80 \%\) confidence interval for \(\mu\).
2. 1. Write down a confidence interval for \(\mu\), based on Jim's sample of 12 students, which has a width of 10 marks.
   2. Determine the percentage confidence level for the interval found in part (b)(i).

1 A factory produces bottles of brown sauce and bottles of tomato sauce.

1. The content, \(Y\) grams, of a bottle of brown sauce is normally distributed with mean \(\mu _ { Y }\) and variance 4. A quality control inspection found that the mean content, \(\bar { y }\) grams, of a random sample of 16 bottles of brown sauce was 450 . Construct a \(95 \%\) confidence interval for \(\mu _ { Y }\).
2. The content, \(X\) grams, of a bottle of tomato sauce is normally distributed with mean \(\mu _ { X }\) and variance \(\sigma ^ { 2 }\). A quality control inspection found that the content, \(x\) grams, of a random sample of 9 bottles of tomato sauce was summarised by $$\sum x = 4950 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 334$$
   1. Construct a 90\% confidence interval for \(\mu _ { X }\).
   2. Holly, the supervisor at the factory, claims that the mean content of a bottle of tomato sauce is 545 grams. Comment, with a justification, on Holly's claim. State the level of significance on which your conclusion is based.
      (3 marks)