5.05d Confidence intervals: using normal distribution

1. Lambs are born in a shed on Mill Farm. The birth weights, \(x \mathrm {~kg}\), of a random sample of 8 newborn lambs are given below.

$$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$

1. Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
2. Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of Mill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
3. Calculate a \(95 \%\) confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.

7. A petrol pump is tested regularly to check that the reading on its gauge is accurate. The random variable \(X\), in litres, is the quantity of petrol actually dispensed when the gauge reads 10.00 litres. \(X\) is known to have distribution \(X \sim \mathrm {~N} \left( \mu , 0.08 ^ { 2 } \right)\)

1. Eight random tests gave the following values of \(x\) $$\begin{array} { l l l l l l l l } 10.01 & 9.97 & 9.93 & 9.99 & 9.90 & 9.95 & 10.13 & 9.94 \end{array}$$
   1. Find a 95\% confidence interval for \(\mu\) to 2 decimal places.
   2. Use your result to comment on the accuracy of the petrol gauge.
2. A sample mean of 9.96 litres was obtained from a random sample of \(n\) tests. A \(90 \%\) confidence interval for \(\mu\) gave an upper limit of less than 10.00 litres. Find the minimum value of \(n\).

6. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population with mean \(\mu\).

1. Show that \(\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)\) is an unbiased estimator of the population mean \(\mu\). A company produces small jars of coffee. Five jars of coffee were taken at random and weighed. The weights, in grams, were as follows $$\begin{array} { l l l l l } 197 & 203 & 205 & 201 & 195 \end{array}$$
2. Calculate unbiased estimates of the population mean and variance of the weights of the jars produced by the company. It is known from previous results that the weights are normally distributed with standard deviation 4.8 g . The manager is going to take a second random sample. He wishes to ensure that there is at least a \(95 \%\) probability that the estimate of the population mean is within 1.25 g of its true value.
3. Find the minimum sample size required.

The weights of bags of rice, \(X \mathrm {~kg}\), have a normal distribution with unknown mean \(\mu \mathrm { kg }\) and known standard deviation \(\sigma \mathrm { kg }\). A random sample of 100 bags of rice gave a \(90 \%\) confidence interval for \(\mu\) of \(( 0.4633,0.5127 )\).

1. Without carrying out any further calculations, use this confidence interval to test whether or not \(\mu = 0.5\) State your hypotheses clearly and write down the significance level you have used. A second random sample, of 150 of these bags of rice, had a mean weight of 0.479 kg .
2. Calculate a \(95 \%\) confidence interval for \(\mu\) based on this second sample.

5. Paul takes the company bus to work. According to the bus timetable he should arrive at work at 0831. Paul believes the bus is not reliable and often arrives late. Paul decides to test the arrival time of the bus and carries out a survey. He records the values of the random variable $$X = \text { number of minutes after } 0831 \text { when the bus arrives. }$$ His results are summarised below. $$n = 15 \quad \sum x = 60 \quad \sum x ^ { 2 } = 1946$$

1. Calculate unbiased estimates of the mean, \(\mu\), and the variance of \(X\). Using the mean of Paul's sample and given \(X \sim \mathrm {~N} \left( \mu , 10 ^ { 2 } \right)\)
2. 1. calculate a 95\% confidence interval for the mean arrival time at work for this company bus.
   2. State an assumption you made about the values in the sample obtained by Paul.
3. Comment on Paul's belief. Justify your answer.

1. The waiting times, in minutes, of patients at a doctor's surgery follows a normal distribution with unknown mean \(\mu\) and known standard deviation \(\sigma\)

A random sample of 120 patients was taken.

1. Find, in the form \(k \sigma\), the width of a \(99 \%\) confidence interval for \(\mu\) based on this sample. Give the value of \(k\) to 2 decimal places. A further random sample of 100 patients from the surgery gave a \(90 \%\) confidence interval for \(\mu\) of \(( 5.14,6.25 )\)
2. Use this confidence interval to determine whether or not it provides evidence that \(\mu = 6\) State the hypotheses being tested here and write down the significance level being used. You do not need to carry out any further calculations.
3. Find the value of \(\sigma\)

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.

6

1. The time taken, in minutes, by Domesat to install a domestic satellite system may be modelled by a normal distribution with unknown mean, \(\mu\), and standard deviation 15 . The times taken, in minutes, for a random sample of 10 installations are as follows. \(\begin{array} { l l l l l l l l l l } 47 & 39 & 25 & 51 & 47 & 36 & 63 & 41 & 78 & 43 \end{array}\) Construct a \(98 \%\) confidence interval for \(\mu\).
2. The time taken, \(Y\) minutes, by Teleair to erect a TV aerial and then connect it to a TV is known to have a mean of 108 and a standard deviation of 28. Estimate the probability that the mean of a random sample of 40 observations of \(Y\) is more than 120 .
3. Indicate, with a reason, where, if at all, in this question you made use of the Central Limit Theorem.
   (2 marks)
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7 An ambulance control centre responds to emergency calls in a rural area. The response time, \(T\) minutes, is defined as the time between the answering of an emergency call at the centre and the arrival of an ambulance at the given location of the emergency. Response times have an unknown mean \(\mu _ { T }\) and an unknown variance.
Anita, the centre's manager, asked Peng, a student on supervised work experience, to record and summarise the values of \(T\) obtained from a random sample of 80 emergency calls. Peng's summarised results were $$\text { Mean, } \bar { t } = 6.31 \quad \text { Variance (unbiased estimate), } s ^ { 2 } = 19.3$$ Only 1 of the 80 values of \(T\) exceeded 20

1. Anita then asked Peng to determine a confidence interval for \(\mu _ { T }\). Peng replied that, from his summarised results, \(T\) was not normally distributed and so a valid confidence interval for \(\mu _ { T }\) could not be constructed.
   1. Explain, using the value of \(\bar { t } - 2 s\), why Peng's conclusion that \(T\) was not normally distributed was likely to be correct.
   2. Explain why Peng's conclusion that a valid confidence interval for \(\mu _ { T }\) could not be constructed was incorrect.
2. Construct a \(98 \%\) confidence interval for \(\mu _ { T }\).
3. Anita had two targets for \(T\). These were that \(\mu _ { T } < 8\) and that \(\mathrm { P } ( T \leqslant 20 ) > 95 \%\). Indicate, with justification, whether each of these two targets was likely to have been met.
   \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-19_2484_1707_223_155}

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]
      
      \includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-28_2488_1728_219_141}

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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3 The time, \(T\) minutes, that parents have to wait before seeing a mathematics teacher at a school parents' evening can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). At a recent parents' evening, a random sample of 9 parents was asked to record the times that they waited before seeing a mathematics teacher. The times, in minutes, are $$\begin{array} { l l l l l l l l l } 5 & 12 & 10 & 8 & 7 & 6 & 9 & 7 & 8 \end{array}$$

1. Construct a \(90 \%\) confidence interval for \(\mu\).
2. Comment on the headteacher's claim that the mean time that parents have to wait before seeing a mathematics teacher is 5 minutes.

1 Alan's journey time, in minutes, to travel home from work each day is known to be normally distributed with mean \(\mu\). Alan records his journey time, in minutes, on a random sample of 8 days as being $$\begin{array} { l l l l l l l l } 36 & 38 & 39 & 40 & 50 & 35 & 36 & 42 \end{array}$$ Construct a \(95 \%\) confidence interval for \(\mu\).

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)

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)

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.