Unknown variance (t-distribution)

1 There are 18 people in Millie's class. To choose a person at random she numbers the people in the class from 1 to 18 and presses the random number button on her calculator to obtain a 3-digit decimal. Millie then multiplies the first digit in this decimal by two and chooses the person corresponding to this new number. Decimals in which the first digit is zero are ignored.

1. Give a reason why this is not a satisfactory method of choosing a person. Millie obtained a random sample of 5 people of her own age by a satisfactory sampling method and found that their heights in metres were \(1.66,1.68,1.54,1.65\) and 1.57 . Heights are known to be normally distributed with variance \(0.0052 \mathrm {~m} ^ { 2 }\).
2. Find a \(98 \%\) confidence interval for the mean height of people of Millie's age.

1 The times taken by members of a large quiz club to complete a challenge have a normal distribution with mean \(\mu\) minutes. The times, \(x\) minutes, are recorded for a random sample of 8 members of the club. The results are summarised as follows, where \(\bar { x }\) is the sample mean. $$\bar { x } = 33.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 94.5$$ Find a 95\% confidence interval for \(\mu\).

1 The lengths of the leaves of a particular type of tree are normally distributed with mean \(\mu \mathrm { cm }\). The lengths, \(x \mathrm {~cm}\), of a random sample of 12 leaves of this type are recorded. The results are summarised as follows. $$\sum x = 91.2 \quad \sum x ^ { 2 } = 695.8$$ Find a 95\% confidence interval for \(\mu\).

1 The times taken by members of a large cycling club to complete a cross-country circuit have a normal distribution with mean \(\mu\) minutes. The times taken, \(x\) minutes, are recorded for a random sample of 14 members of the club. The results are summarised as follows, where \(\bar { x }\) is the sample mean. $$\bar { x } = 42.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 941.5$$ Find a 95\% confidence interval for \(\mu\).

1 A scientist is investigating the lengths of the leaves of a certain type of plant. The scientist assumes that the lengths of the leaves of this type of plant are normally distributed. He measures the lengths, \(x \mathrm {~cm}\), of the leaves of a random sample of 8 plants of this type. His results are as follows. \(\begin{array} { l l l l l l l l } 3.5 & 4.2 & 3.8 & 5.2 & 2.9 & 3.7 & 4.1 & 3.2 \end{array}\) Find a \(90 \%\) confidence interval for the population mean length of leaves of this type of plant.

4 A set of bathroom scales is known to operate with an error which is normally distributed. One morning a man weighs himself 4 times. The 4 values for his mass, in kg , which can be considered to be a random sample are as follows. $$\begin{array} { l l l l } 62.6 & 62.8 & 62.1 & 62.5 \end{array}$$

1. Find a \(95 \%\) confidence interval for his mass. Give the end-points of the interval correct to 3 decimal places.
2. Based on these results, a \(y \%\) confidence interval has width 0.482 . Find \(y\).

3 A machine produces circular metal discs whose radii have a normal distribution with mean \(\mu \mathrm { cm }\). A random sample of five discs is selected and their radii, in cm, are as follows. $$\begin{array} { l l l l l } 6.47 & 6.52 & 6.46 & 6.47 & 6.51 \end{array}$$

1. Calculate a \(95 \%\) confidence interval for \(\mu\).
2. Hence state a 95\% confidence interval for the mean circumference of a disc.

6 The amount of caffeine in a randomly selected cup of coffee dispensed by a machine has a normal distribution. The amount of caffeine in each of a random sample of 25 cups was measured. The sample mean was 110.4 mg and the unbiased estimate of the population variance was \(50.42 \mathrm { mg } ^ { 2 }\). Calculate a 90\% confidence interval for the mean amount of caffeine dispensed.

6 A machine produces metal discs whose diameters have a normal distribution. The mean of this distribution is intended to be 10 cm . Accuracy is checked by measuring the diameters of a random sample of six discs. The diameters, in cm, are as follows. $$\begin{array} { l l l l l l } 10.03 & 10.02 & 9.98 & 10.06 & 10.08 & 10.01 \end{array}$$ Calculate a 99\% confidence interval for the mean diameter of all discs produced by the machine. Deduce a 99\% confidence interval for the mean circumference of all discs produced by the machine.

7 The speed \(v\) at which a javelin is thrown by an athlete is measured in \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results for 10 randomly chosen throws are summarised by $$\Sigma v = 1110.8 , \quad \Sigma ( v - \bar { v } ) ^ { 2 } = 333.9$$ where \(\bar { v }\) is the sample mean.

1. Stating any necessary assumption, calculate a \(99 \%\) confidence interval for the mean speed of a throw. The results for a further 5 randomly chosen throws are now combined with the above results. It is found that the sample variance is smaller than that used in part (i).
2. State, with reasons, whether a \(95 \%\) confidence interval calculated from the combined 15 results will be wider or less wide than that found in part (i).

5 A random sample of 10 observations of a normal variable \(X\) gave the following summarised data, where \(\bar { x }\) is the sample mean. $$\Sigma x = 222.8 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 4.12$$ Find a 95\% confidence interval for the population mean.

6 The heights, in metres, of a random sample of 8 trees of a particular type are as follows.
8.4
9.2
10.8
11.3
11.5
12.8
12.1
14.2 Assuming that heights of trees of this type are normally distributed, calculate a \(95 \%\) confidence interval for the mean height of trees of this type.

5 A random sample of 10 observations of a normal variable \(X\) gave the following summarised data, where \(\bar { x }\) is the sample mean. $$\Sigma x = 222.8 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 4.12$$ Find a \(95 \%\) confidence interval for the population mean.

7 A club secretary collects data about the time, \(T\) minutes, needed to process the details of a new member. The mean of \(T\) is denoted by \(\mu\). The variance of \(T\) is denoted by \(\sigma ^ { 2 }\). The results of a random sample of 40 observations of \(T\) are summarised as follows. \(\mathrm { n } = 40 \quad \Sigma \mathrm { t } = 396.0 \quad \Sigma \mathrm { t } ^ { 2 } = 4271.40\)

1. Determine a 99\% confidence interval for \(\mu\).
2. The secretary discovers that over a long period the value of \(\sigma ^ { 2 }\) is in fact 10.0 . The secretary collects an independent random sample of 50 observations of \(T\) and constructs a new 99\% confidence interval for \(\mu\) based on this sample of size 50 , but using \(\sigma ^ { 2 } = 10.0\). Find the probability that this new confidence interval contains the value \(\mu + 1.6\).

7. A machine produces metal containers. The weights of the containers are normally distributed. A random sample of 10 containers from the production line was weighed, to the nearest 0.1 kg , and gave the following results $$\begin{array} { l l l l l } 49.7 , & 50.3 , & 51.0 , & 49.5 , & 49.9 \\ 50.1 , & 50.2 , & 50.0 , & 49.6 , & 49.7 . \end{array}$$

1. Find unbiased estimates of the mean and variance of the weights of the population of metal containers. The machine is set to produce metal containers whose weights have a population standard deviation of 0.5 kg .
2. Estimate the limits between which \(95 \%\) of the weights of metal containers lie.
3. Determine the \(99 \%\) confidence interval for the mean weight of metal containers.

8. Observations have been made over many years of \(T\), the noon temperature in \({ } ^ { \circ } \mathrm { C }\), on 21 st March at Sunnymere. The records for a random sample of 12 years are given below.
5.2, 3.1, 10.6, 12.4, 4.6, 8.7, 2.5, 15.3, -1.5, 1.8, 13.2, 9.3.

1. Find unbiased estimates of the mean and variance of \(T\). Over the years, the standard deviation of \(T\) has been found to be 5.1.
2. Assuming a normal distribution find a \(90 \%\) confidence interval for the mean of \(T\).
   (5) A meteorologist claims that the mean temperature at noon in Sunnymere on 21st March is \(4 ^ { \circ } \mathrm { C }\).
3. Use your interval to comment on the meteorologist's claim.

7 Customers buying euros ( €) at a travel agency must pay for them in pounds ( \(\pounds\) ). The amounts paid, \(\pounds x\), by a sample of 40 customers were, in ascending order, as follows.

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 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?