5.05d Confidence intervals: using normal distribution

4 A machine is used to fill 5-litre plastic containers with vinegar. The volume, in litres, of vinegar in a container filled by the machine may be assumed to be normally distributed with mean \(\mu\) and standard deviation 0.08 . A quality control inspector requires a \(99 \%\) confidence interval for \(\mu\) to be constructed such that it has a width of at most 0.05 litres. Calculate, to the nearest 5, the sample size necessary in order to achieve the inspector's requirement.

4 The waiting times for patients to see a doctor in a hospital can be modelled with a normal distribution with known variance of 10 minutes. 4

1. A random sample of 100 patients has a total waiting time of 3540 minutes.
   Calculate a \(98 \%\) confidence interval for the population mean of waiting times, giving values to four significant figures.
   4
2. Dante conducts a hypothesis test with the sample from part (a) on the waiting times. Dante's hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 38 \\ & \mathrm { H } _ { 1 } : \mu \neq 38 \end{aligned}$$ Dante uses a \(2 \%\) level of significance.
   Explain whether Dante accepts or rejects the null hypothesis.

3 Fiona is studying the heights of corn plants on a farm. She measures the height, \(x \mathrm {~cm}\), of a random sample of 200 corn plants on the farm.
The summarised results are as follows: $$\sum x = 60255 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 995$$ Calculate a \(96 \%\) confidence interval for the population mean of heights of corn plants on the farm, giving your values to one decimal place.

4 The height of lilac trees, in metres, can be modelled by a normal distribution with variance 0.7 A random sample of \(n\) lilac trees is taken and used to construct a 99\% confidence interval for the population mean. This confidence interval is \(( 5.239,5.429 )\) 4

1. Find the value of \(n\) 4
2. Joey claims that the mean height of lilac trees is 5.3 metres.
   State, with a reason, whether the confidence interval supports Joey's claim.

5 Rebekah is investigating the distances, \(X\) light years, between the Earth and visible stars in the night sky. She determines the distance between the Earth and a star for a random sample of 100 visible stars. The summarised results are as follows: $$\sum x = 35522 \quad \text { and } \quad \sum x ^ { 2 } = 32902257$$ 5

1. Calculate a 97\% confidence interval for the population mean of \(X\), giving your values to the nearest light year.
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2. Mike claims that the population mean is 267 light years. Rebekah says that the confidence interval supports Mike's claim. State, with a reason, whether Rebekah is correct.

3 The random variable \(X\) has a normal distribution with known variance 15.7 A random sample of size 120 is taken from \(X\) The sample mean is 68.2 Find a 94\% confidence interval for the population mean of \(X\) Give your limits to three significant figures.

2 Amy takes a sample of size 50 from a normal distribution with mean \(\mu\) and variance 16 She conducts a hypothesis test with hypotheses: $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 52 \\ & \mathrm { H } _ { 1 } : \mu > 52 \end{aligned}$$ She rejects the null hypothesis if her sample has a mean greater than 53
The actual population mean is 53.5
Find the probability that Amy makes a Type II error.
Circle your answer. \(0.4 \% 3.9 \% 18.9 \% 15.0 \%\)

3 Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes. 3

1. Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place.
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2. Alan claims that his mean journey time to work is 30 minutes.
   State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim.
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3. Suppose that the standard deviation is not known but a sample standard deviation is found from Alan's sample and calculated to be 6 Explain how the working in part (a) would change.

3 The mass of male giraffes is assumed to have a normal distribution. Duncan takes a random sample of 600 male giraffes.
The mean mass of the sample is 1196 kilograms.
The standard deviation of the sample is 98 kilograms.
3

1. Construct a 94\% confidence interval for the mean mass of male giraffes, giving your values to one decimal place.
   3
2. Explain whether or not your answer to part (a) would change if a sample of size 5 was taken with the same mean and standard deviation.

4 Oscar is studying the daily maximum temperature in \({ } ^ { \circ } \mathrm { C }\) in a village during the month of June. He constructs a \(95 \%\) confidence interval of width \(0.8 ^ { \circ } \mathrm { C }\) using a random sample of 150 days. He assumes that the daily maximum temperature has a normal distribution.
4

1. Find the standard deviation of Oscar's sample, giving your answer to three significant figures.
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2. Oscar calculates the mean of his sample to be \(25.3 ^ { \circ } \mathrm { C }\) He claims that the population mean is \(26.0 ^ { \circ } \mathrm { C }\) Explain whether or not his confidence interval supports his claim.
   4
3. Explain how Oscar could reduce the width of his 95\% confidence interval.

5 The mass, \(X\), in grams of a particular type of apple is modelled using a normal distribution. A random sample of 12 apples is collected and the summarised results are $$\sum x = 1038 \quad \text { and } \quad \sum x ^ { 2 } = 90100$$ 5

1. A 99\% confidence interval for the population mean of the masses of the apples is constructed using the random sample. Show that the confidence interval is \(( 81.7,91.3 )\) with values correct to three significant figures.
   5
2. Padraig claims that the population mean mass of the apples is 85 grams. He carries out a hypothesis test at the \(1 \%\) level of significance using the random sample of 12 apples. The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 85 \\ & \mathrm { H } _ { 1 } : \mu \neq 85 \end{aligned}$$ State, with a reason, whether the null hypothesis is accepted or rejected.
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3. Interpret, in context, the conclusion to the hypothesis test in part (b).

3 The masses of tins of a particular brand of spaghetti are normally distributed with mean \(\mu\) grams and standard deviation 4.1 grams. A random sample of 11 tins of spaghetti has a mean mass of 401.8 grams.
Construct a \(98 \%\) confidence interval for \(\mu\), giving your values to one decimal place.

10 The label on a particular size of milk carton states that it contains 1.5 litres of milk. In an investigation at the packaging plant the contents, \(x\) litres, of each of 60 randomly selected cartons are measured. The data are summarised as follows. $$\Sigma x = 89.758 \quad \Sigma x ^ { 2 } = 134.280$$

1. Estimate the variance of the underlying population.
2. Find a 95\% confidence interval for the mean of the underlying population.
3. What does the confidence interval which you have calculated suggest about the statement on the carton? Each day for 300 days a random sample of 60 cartons is selected and for each sample a \(95 \%\) confidence interval is constructed.
4. Explain why the confidence intervals will not be identical.
5. What is the expected number of confidence intervals to contain the population mean?

1. Rachel records the times taken, in minutes, to cycle into town from her house on a random sample of 10 days. Her results are shown below.

$$\begin{array} { l l l l l l l l l l } 15 \cdot 5 & 14 \cdot 9 & 16 \cdot 2 & 17 \cdot 3 & 14 \cdot 8 & 14 \cdot 2 & 16 \cdot 0 & 14 \cdot 2 & 15 \cdot 5 & 15 \cdot 1 \end{array}$$ Assuming that these data come from a normal distribution with mean \(\mu\) and variance \(0 \cdot 9\), calculate a \(90 \%\) confidence interval for \(\mu\).

4. The Department of Health recommends that adults aged 18 to 65 should take part in at least 150 minutes of aerobic exercise per week. The results of a survey show that 940 out of 2000 randomly selected adults aged 18 to 65 in Wales take part in at least 150 minutes of aerobic exercise per week.

1. Calculate an approximate \(95 \%\) confidence interval for the proportion of adults aged 18 to 65 in Wales who take part in at least 150 minutes of aerobic exercise per week.
2. Give two reasons why the interval is approximate.
3. Suppose that a \(99 \%\) confidence interval is required, and that the width of the interval is to be no greater than \(0 \cdot 04\). Estimate the minimum additional number of adults to be surveyed to satisfy this requirement.

1
The continuous random variable \(X\) has the distribution \(\mathrm { N } ( \mu , 30 )\). The mean of a random sample of 8 observations of \(X\) is 53.1 . Determine a \(95 \%\) confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures.

8 Two groups of Year 12 pupils, one at each of schools \(A\) and \(B\), are given the same mathematics test. The scores, \(x\) and \(y\), of pupils at schools \(A\) and \(B\) respectively are summarised as follows.

1. Assuming that the two groups are random samples from independent normal populations with means \(\mu _ { A }\) and \(\mu _ { B }\) respectively and a common, but unknown, variance, construct a \(98 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\).
2. Comment, with a reason, on any difference in ability between the two schools.

3 The fuel economy of two similar cars produced by manufacturers \(A\) and \(B\) was compared. A random sample of 15 cars was selected from manufacturer \(A\) and a random sample of 10 cars was selected from manufacturer \(B\). All the selected cars were driven over the same distance and the petrol consumption in miles per gallon (mpg) was calculated for each car. The results, \(x _ { A } \operatorname { mpg }\) and \(x _ { B } \operatorname { mpg }\) for cars from manufacturers \(A\) and \(B\) respectively, are summarised below, where \(\bar { x }\) denotes the sample mean and \(n\) the sample size. $$\begin{array} { l l l } \Sigma x _ { A } = 460.5 & \Sigma \left( x _ { A } - \bar { x } _ { A } \right) ^ { 2 } = 156.88 & n _ { A } = 15 \\ \Sigma x _ { B } = 334 & \Sigma \left( x _ { B } - \bar { x } _ { B } \right) ^ { 2 } = 123.97 & n _ { B } = 10 \end{array}$$

1. (a) Assuming that the populations are normally distributed with a common variance, show that the pooled estimate of this common variance is 12.21 , correct to 4 significant figures. [2]
   (b) Construct a 95\% confidence interval for \(\mu _ { B } - \mu _ { A }\), the difference in the population means for manufacturers \(A\) and \(B\).
2. Comment on a claim that the fuel economy for manufacturer \(B\) 's cars is better than that for manufacturer \(A\) 's cars.

1. A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { \theta } \mathrm { e } ^ { - \frac { x } { \theta } } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ where \(\theta\) is a positive constant. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
2. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population with the distribution in part (i). The estimator \(T\) is defined by \(T = k \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }\), where \(k\) is a constant. Find the value of \(k\) such that \(T\) is an unbiased estimator of \(\theta ^ { 2 }\).

1. The discrete random variable \(X\) has distribution \(\operatorname { Geo } ( p )\). Show that the moment generating function of \(X\) is given by \(\mathrm { M } _ { X } ( t ) = \frac { p \mathrm { e } ^ { t } } { 1 - q \mathrm { e } ^ { t } }\), where \(q = 1 - p\).
2. Use the moment generating function to find
   1. \(\mathrm { E } ( X )\),
   2. \(\operatorname { Var } ( X )\).
   3. An unbiased six-sided die is thrown repeatedly until a five is obtained, and \(Y\) denotes the number of throws up to and including the throw on which the five is obtained. Find \(\mathrm { P } ( | Y - \mu | < \sigma )\), where \(\mu\) and \(\sigma\) are the mean and standard deviation, respectively, of the distribution of \(Y\).
   1. The continuous random variable \(X\) has a uniform distribution over the interval \(0 < x < \frac { 1 } { 2 } \pi\). Show that the probability density function of \(Y\), where \(Y = \sin X\), is given by $$\mathrm { f } ( y ) = \begin{cases} \frac { 2 } { \pi \sqrt { 1 - y ^ { 2 } } } & 0 < y < 1 \\ 0 & \text { otherwise. } \end{cases}$$
   2. Deduce, using the probability density function, the exact values of
   
   (a) the median value of \(Y\),
   (b) \(\mathrm { E } ( Y )\).

3 Small amounts of a potentially hazardous chemical are discharged into a river from a nearby industrial site. A random sample of size 6 was taken from the river and the concentration of the chemical present in each item was measured in grams per litre. The results are shown below. $$\begin{array} { l l l l l l } 1.64 & 1.53 & 1.78 & 1.60 & 1.73 & 1.77 \end{array}$$

1. Assuming that the sample was taken from a normal distribution with known variance 0.01 , construct a \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
2. If instead the sample was taken from a normal distribution, but with unknown variance, construct a revised \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
3. If the mean concentration of the chemical in the river exceeds 1.8 grams per litre, then remedial action needs to be taken. Comment briefly on the need for remedial action in the light of the results in parts (i) and (ii).

4 The broadband speed in village \(P\) was measured on 8 randomly selected occasions and the broadband speed in village \(Q\) was measured on 6 randomly selected occasions. The results, measured in megabits per second, are shown below.

1. Calculate a \(90 \%\) confidence interval for the difference in mean broadband speed in these two villages.
2. State two assumptions that you have made in carrying out the calculation.

5 The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\), where \(p > 0.5\). A random sample of \(4 n\) observations of \(X\) is taken and \(\bar { X }\) denotes the sample mean. It is given that \(\mathrm { E } ( \bar { X } ) = 180\) and \(\operatorname { Var } ( \bar { X } ) = 0.0225\).

1. Find
   1. the values of \(p\) and \(n\),
   2. \(\mathrm { P } ( \bar { X } < 179.8 )\),
   3. the value of \(a\) for which \(\mathrm { P } ( 180 - a < \bar { X } < 180 + a ) = 0.99\), giving your answer correct to 2 decimal places.
   4. State how you have used the Central Limit Theorem in part (i).

2 The pH value, \(X\), which is a measure of acidity, was measured for soil taken from a random sample of 20 villages in which rhododendrons grow well. The results are summarised below, where \(\bar { x }\) denotes the sample mean. You may assume that the sample is selected from a normal population. $$\Sigma x = 114 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 2.382$$

1. Calculate a \(98 \%\) confidence interval for the mean pH value in villages where rhododendrons grow well, giving 3 decimal places in your answer.
2. Comment, justifying your answer, on a suggestion that the average pH value in villages where rhododendrons grow well is 5.5.

1 An investigation was carried out of the lengths of commuters' journeys. For a random sample of 500 commuters, the mean journey time was 75 minutes, and the standard deviation was 40 minutes.

1. Calculate a 95\% confidence interval for the mean journey time.
2. Explain whether you need to assume that journey times are normally distributed.

1

1. Explain the meaning of the term ' \(95 \%\) confidence interval'.
2. The values of five independent observations of a normally distributed random variable are as follows. $$\begin{array} { l l l l l } 35.2 & 38.2 & 39.7 & 41.6 & 43.9 \end{array}$$ Obtain a 95\% confidence interval for the population mean.

3 In a random sample of 100 voters from a constituency, 32 said that they would support the Cyan Party.

1. Find an approximate \(99 \%\) confidence interval for the proportion of voters in the constituency who would support the Cyan Party.
2. Using the given sample proportion, estimate the smallest size of sample needed for the width of a \(99 \%\) confidence interval to be less than 0.04 .