5.05d Confidence intervals: using normal distribution

1 Dimitra is an athlete who competes in 400 m races. The times, in seconds, for her first six races of the 2012 season were $$\begin{array} { l l l l l l } 54.86 & 53.09 & 53.75 & 52.88 & 51.97 & 51.81 \end{array}$$

1. Assuming that these data form a random sample from a normal distribution, construct a \(95 \%\) confidence interval for the mean time of Dimitra's races in the 2012 season, giving the limits to two decimal places.
2. For the 2011 season, Dimitra's mean time for her races was 53.41 seconds. After her first six races of the 2012 season, her coach claimed that the data showed that she would be more successful in races during the 2012 season than during the 2011 season. Make two comments about the coach's claim.

3 The heights, in metres, of a random sample of 10 students attending Higrade School are recorded below. \(\begin{array} { l l l l l l l l l } 1.76 & 1.59 & 1.54 & 1.62 & 1.49 & 1.52 & 1.56 & 1.47 & 1.75 \end{array} 1.50\) Assume that the heights of students attending Higrade School are normally distributed.

1. Calculate unbiased estimates for the mean and variance of the heights of students attending Higrade School.
   (3 marks)
2. Construct a 90\% confidence interval for the mean height of students attending Higrade School.
   (5 marks)

2 The weights of lions kept in captivity at Wildcat Safari Park are normally distributed.
The weights, in kilograms, of a random sample of five lions were recorded as $$\begin{array} { l l l l l } 46 & 48 & 57 & 49 & 54 \end{array}$$

1. Construct a 95\% confidence interval for the mean weight of lions kept in captivity at Wildcat Safari Park.
2. State the probability that this confidence interval does not contain the mean weight of lions kept in captivity at Wildcat Safari Park.

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?

2

1. The continuous random variable \(X\) has a rectangular distribution defined by the probability density function $$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u \\ 0 & \text { otherwise } \end{cases}$$ where \(u\) is a constant.
   1. Show that \(u = \frac { 10 } { \pi }\).
   2. Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of \(\pi\), values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   3. Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length \(\left( X + \frac { 10 } { \pi } \right)\).
2. A machine produces circular discs which have an area of \(Y \mathrm {~cm} ^ { 2 }\). The distribution of \(Y\) has mean \(\mu\) and variance 25 . A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be \(40.5 \mathrm {~cm} ^ { 2 }\). Calculate a 95\% confidence interval for \(\mu\). Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area. The table shows Emily's collected data, denoted by \(O _ { i }\), together with the corresponding expected frequencies, \(E _ { i }\), necessary for a \(\chi ^ { 2 }\) test.

   \multirow{2}{*}{}	\(\boldsymbol { \geq } \mathbf { 5 }\) GCSEs		\(\mathbf { 1 } \boldsymbol { \leqslant }\) GCSEs < \(\mathbf { 5 }\)		No GCSEs
   	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)
   Jolliffe College for the Arts	187	193.15	93	90.62	30	26.23
   Volpe Science Academy	175	184.43	97	86.52	24	25.05
   Radok Music School	183	183.81	78	86.23	34	24.96
   Bailey Language School	265	248.61	112	116.63	22	33.76

   Emily used these values to correctly conduct a \(\chi ^ { 2 }\) test at the \(1 \%\) level of significance.

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)

1 At the start of the 2012 season, the ages of the members of the Warwickshire Acorns Cricket Club could be modelled by a normal random variable, \(X\) years, with mean \(\mu\) and standard deviation \(\sigma\). The ages, \(x\) years, of a random sample of 15 such members are summarised below. $$\sum x = 546 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1407.6$$

1. Construct a \(98 \%\) confidence interval for \(\mu\), giving the limits to one decimal place.
   (6 marks)
2. At the start of the 2005 season, the mean age of the members was 40.0 years. Use your confidence interval constructed in part (a) to indicate, with a reason, whether the mean age had changed.

1 Gemma, a biologist, studies guillemots, which are a species of seabird. She has found that the weight of an adult guillemot may be modelled by a normal distribution with mean \(\mu\) grams. During 2012, she measured the weight, \(x\) grams, of each of a random sample of 9 adult guillemots and obtained the following results. $$\sum x = 8532 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 38538$$

1. Construct a 98\% confidence interval for \(\mu\) based on these data.
2. The corresponding confidence interval for \(\mu\) obtained by Gemma based on a random sample of 9 adult guillemots measured during 2011 was \(( 927,1063 )\), correct to the nearest gram.
   1. Find the mean weight of guillemots in this sample.
   2. Studies of some other species of seabird have suggested that their mean weights were less in 2012 than in 2011. Comment on whether Gemma's two confidence intervals provide evidence that the mean weight of guillemots was less in 2012 than in 2011.
      (2 marks)

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).

1 Vanya collected five samples of air and measured the carbon dioxide content of each sample, in parts per million by volume (ppmv). The results were as follows. $$\begin{array} { l l l l l } 387 & 375 & 382 & 379 & 381 \end{array}$$

1. Assuming that these data form a random sample from a normal distribution with mean \(\mu\) ppmv, construct a \(90 \%\) confidence interval for \(\mu\).
   [0pt] [6 marks]
2. Vanya repeated her sampling procedure on each of 30 days and, for each day's results, a \(90 \%\) confidence interval for \(\mu\) was constructed. On how many of these 30 days would you expect \(\mu\) to lie outside that day's confidence interval?
   [0pt] [1 mark]

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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3 A machine fills bags with frozen peas. Measurements taken over several weeks have shown that the standard deviation of the weights of the filled bags of peas has been 2.2 grams. Following maintenance on the machine, a quality control inspector selected 8 bags of peas. The weights, in grams, of the bags were $$\begin{array} { l l l l l l l l } 910.4 & 908.7 & 907.2 & 913.2 & 905.6 & 911.1 & 909.5 & 907.9 \end{array}$$ It may be assumed that the bags constitute a random sample from a normal distribution.

1. Giving the limits to four significant figures, calculate a 95\% confidence interval for the mean weight of a bag of frozen peas filled by the machine following the maintenance:
   1. assuming that the standard deviation of the weights of the bags of peas is known to be 2.2 grams;
   2. assuming that the standard deviation of the weights of the bags of peas may no longer be 2.2 grams.
2. The weight printed on the bags of peas is 907 grams. One of the inspector's concerns is that bags should not be underweight. Make two comments about this concern with regard to the data and your calculated confidence intervals.
   [0pt] [2 marks]

2 A survey of a random sample of 200 passengers on UK internal flights revealed that 132 of them were on business trips.

1. Construct an approximate \(98 \%\) confidence interval for the proportion of passengers on UK internal flights that are on business trips.
2. Hence comment on the claim that more than 60 per cent of passengers on UK internal flights are on business trips.

5 The daily number of emergency calls received from district A may be modelled by a Poisson distribution with a mean of \(\lambda _ { \mathrm { A } }\). The daily number of emergency calls received from district B may be modelled by a Poisson distribution with a mean of \(\lambda _ { \mathrm { B } }\). During a period of 184 days, the number of emergency calls received from district A was 3312, whilst the number received from district B was 2760.

1. Construct an approximate \(95 \%\) confidence interval for \(\lambda _ { \mathrm { A } } - \lambda _ { \mathrm { B } }\).
2. State one assumption that is necessary in order to construct the confidence interval in part (a).

1 An analysis of a random sample of 150 urban dwellings for sale showed that 102 are semi-detached. An analysis of an independent random sample of 80 rural dwellings for sale showed that 36 are semi-detached.

1. Construct an approximate \(99 \%\) confidence interval for the difference between the proportion of urban dwellings for sale that are semi-detached and the proportion of rural dwellings for sale that are semi-detached.
2. Hence comment on the claim that there is no difference between these two proportions.

3 The proportion, \(p\), of an island's population with blood type \(\mathrm { A } \mathrm { Rh } ^ { + }\)is believed to be approximately 0.35 . A medical organisation, requiring a more accurate estimate, specifies that a \(98 \%\) confidence interval for \(p\) should have a width of at most 0.1 . Calculate, to the nearest 10, an estimate of the minimum sample size necessary in order to achieve the organisation's requirement.

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.
   \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-05_2484_1709_223_153}

2 The number of emergency calls received by a fire station may be modelled by a Poisson distribution. During a given period of 13 weeks, the station received a total of 108 emergency calls.

1. Construct an approximate \(98 \%\) confidence interval for the average weekly number of emergency calls received by the station.
2. Hence comment on the station officer's claim that the station receives an average of one emergency call per day.
   (2 marks)

4
The waiting time at a hospital's A\&E department may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\frac { \mu } { 2 }\).
The department's manager wishes a \(95 \%\) confidence interval for \(\mu\) to be constructed such that it has a width of at most \(0.2 \mu\).
Calculate, to the nearest 10, an estimate of the minimum sample size necessary in order to achieve the manager's wish.
(5 marks)
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5
An examination of 160 e-mails received by Gopal showed that 72 had attachments. An examination of 250 e-mails received by Haley showed that 102 had attachments.
Stating two necessary assumptions about the selection of e-mails, construct an approximate \(99 \%\) confidence interval for the difference between the proportion of e-mails received by Gopal that have attachments and the proportion of e-mails received by Haley that have attachments.
(8 marks)

5 A random sample of 125 people was selected from a council's electoral roll. Of these, 68 were in favour of a proposed local building plan.

1. Construct an approximate 98\% confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal.
2. Calculate, to the nearest 5, an estimate of the minimum sample size necessary in order that an approximate \(98 \%\) confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal has a width of at most 10 per cent.

1 The number of telephone calls per hour to an out-of-hours doctors' service may be modelled by a Poisson distribution. The total number of telephone calls received during a random sample of 12 weekday night shifts, all of the same duration, was 392.

1. Calculate an approximate \(98 \%\) confidence interval for the mean number of calls received per weekday night shift.
2. The mean number of calls received during weekend shifts of 48 hours' total duration is 136.8 . Comment on a claim that the mean number of calls per hour during weekend shifts is greater than that during weekday night shifts, which are each of \(\mathbf { 1 4 }\) hours' duration.
   (3 marks)

3 A builders' merchant's depot has two machines, X and Y , each of which can be used for filling bags with sand or gravel. The weight, in kilograms, delivered by machine X may be modelled by a normal distribution with mean \(\mu _ { \mathrm { X } }\) and standard deviation 25 . The weight, in kilograms, delivered by machine Y may be modelled by a normal distribution with mean \(\mu _ { \mathrm { Y } }\) and standard deviation 30 . Fred, the depot's yardman, records the weights, in kilograms, of a random sample of 10 bags of sand delivered by machine X as \(\begin{array} { l l l l l l l l l l } 1055 & 1045 & 1000 & 985 & 1040 & 1025 & 1005 & 1030 & 1015 & 1060 \end{array}\) He also records the weights, in kilograms, of a random sample of 8 bags of gravel delivered by machine Y as $$\begin{array} { l l l l l l l l } 1085 & 1055 & 1055 & 1000 & 1035 & 1050 & 1005 & 1075 \end{array}$$

1. Construct a \(95 \%\) confidence interval for \(\mu _ { \mathrm { Y } } - \mu _ { \mathrm { X } }\), giving the limits to the nearest 5 kg .
2. Dot, the depot's manager, commented that Fred's data collection may have been biased. Justify her comment and explain how the possible bias could have been eliminated.
   (2 marks)

1 A hotel's management is concerned about the quality of the free pens that it provides in its meeting rooms. The hotel's assistant manager tests a random sample of 200 such pens and finds that 23 of them fail to write immediately.

1. Calculate an approximate \(\mathbf { 9 6 \% }\) confidence interval for the proportion of pens that fail to write immediately.
2. The supplier of the pens to the hotel claims that at most 2 in 50 pens fail to write immediately. Comment, with numerical justification, on the supplier's claim.
   [0pt] [2 marks] QUESTION
   PART Answer space for question 1

4 A sample of 50 male Eastern Grey kangaroos had a mean weight of 42.6 kg and a standard deviation of 6.2 kg . A sample of 50 male Western Grey kangaroos had a mean weight of 39.7 kg and a standard deviation of 5.3 kg .

1. Construct a 98\% confidence interval for the difference between the mean weight of male Eastern Grey kangaroos and that of male Western Grey kangaroos.
   [0pt] [5 marks]
2. 1. What assumption about the selection of each of the two samples was it necessary to make in order that the confidence interval constructed in part (a) was valid?
      [0pt] [1 mark]
   2. Why was it not necessary to assume anything about the distributions of the weights of male kangaroos in order that the confidence interval constructed in part (a) was valid?
      [0pt] [2 marks]