Single sample confidence interval t-distribution

1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4

5 A large number of children are competing in a throwing competition. The distances, in metres, thrown by a random sample of 8 children are as follows. \(\begin{array} { l l l l l l l l } 19.8 & 22.1 & 24.4 & 21.5 & 20.8 & 26.3 & 23.7 & 25.0 \end{array}\)

1. Assuming that distances are normally distributed, test, at the \(5 \%\) significance level, whether the population mean distance thrown is more than 22.0 metres.
2. Find a 95\% confidence interval for the population mean distance thrown.

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.

5 Raman is researching the heights of male giraffes in a particular region. Raman assumes that the heights of male giraffes in this region are normally distributed. He takes a random sample of 8 male giraffes from the region and measures the height, in metres, of each giraffe. These heights are as follows. $$\begin{array} { c c c c c c c c } 5.2 & 5.8 & 4.9 & 6.1 & 5.5 & 5.9 & 5.4 & 5.6 \end{array}$$

1. Find a \(90 \%\) confidence interval for the population mean height of male giraffes in this region. [5]
   Raman claims that the population mean height of male giraffes in the region is less than 5.9 metres.
2. Test at the \(2.5 \%\) significance level whether this sample provides sufficient evidence to support Raman's claim.

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 machine is set to produce metal discs with mean diameter 15.4 mm . In order to test the correctness of the setting, a random sample of 12 discs was selected and the diameters, \(x \mathrm {~mm}\), were measured. The results are summarised by \(\Sigma x = 177.6\) and \(\Sigma x ^ { 2 } = 2640.40\). Diameters may be assumed to be normally distributed with mean \(\mu \mathrm { mm }\).

1. Find a \(95 \%\) confidence interval for \(\mu\).
2. Test, at the \(5 \%\) significance level, the null hypothesis \(\mu = 15.4\) against the alternative hypothesis \(\mu < 15.4\).

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 \(y\).

4 The time interval, \(T\) minutes, between consecutive stoppages of a particular grinding machine is regularly measured. \(T\) is normally distributed with mean \(\mu\).
24 randomly chosen values of \(T\) are summarised by $$\sum _ { i = 1 } ^ { 24 } t _ { i } = 348.0 \text { and } \sum _ { i = 1 } ^ { 24 } t _ { i } ^ { 2 } = 5195.5 .$$

1. Calculate a symmetric \(95 \%\) confidence interval for \(\mu\).
2. For the machine to be working acceptably, \(\mu\) should be at least 15.0 . Using a test at the 10\% significance level, decide whether the machine is working acceptably.

1 A certain industrial process requires a supply of water. It has been found that, for best results, the mean water pressure in suitable units should be 7.8. The water pressure is monitored by taking measurements at regular intervals. On a particular day, a random sample of the measurements is as follows. $$\begin{array} { l l l l l l l l l } 7.50 & 7.64 & 7.68 & 7.51 & 7.70 & 7.85 & 7.34 & 7.72 & 7.74 \end{array}$$ These data are to be used to carry out a hypothesis test concerning the mean water pressure.

1. Why is a test based on the Normal distribution not appropriate in this case?
2. What distributional assumption is needed for a test based on the \(t\) distribution?
3. Carry out a \(t\) test, with a \(2 \%\) level of significance, to see whether it is reasonable to assume that the mean pressure is 7.8 .
4. Explain what is meant by a \(95 \%\) confidence interval.
5. Find a \(95 \%\) confidence interval for the actual mean water pressure.

1 Gerry runs 5000 -metre races for his local athletics club. His coach has been monitoring his practice times for several months and he believes that they can be modelled using a Normal distribution with mean 15.3 minutes. The coach suggests that Gerry should try running with a pacemaker in order to see if this can improve his times. Subsequently a random sample of 10 of Gerry's times with the pacemaker is collected to see if any reduction has been achieved. The sample of times (in minutes) is as follows. $$\begin{array} { l l l l l l l l l l } 14.86 & 15.00 & 15.62 & 14.44 & 15.27 & 15.64 & 14.58 & 14.30 & 15.08 & 15.08 \end{array}$$

1. Why might a \(t\) test be used for these data?
2. Using a \(5 \%\) significance level, carry out the test to see whether, on average, Gerry's times have been reduced.
3. What is meant by 'a \(5 \%\) significance level'? What would be the consequence of decreasing the significance level?
4. Find a \(95 \%\) confidence interval for the true mean of Gerry's times using a pacemaker.

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.

8 A large number of long jumpers are competing in a national long jump competition. The distances, in metres, jumped by a random sample of 7 competitors are as follows. $$\begin{array} { l l l l l l l } 6.25 & 7.01 & 5.74 & 6.89 & 7.24 & 5.64 & 6.52 \end{array}$$ Assuming that distances are normally distributed, test, at the \(5 \%\) significance level, whether the mean distance jumped by long jumpers in this competition is greater than 6.2 metres. The distances jumped by another random sample of 8 long jumpers in this competition are recorded. Using the data from this sample of 8 long jumpers, a \(95 \%\) confidence interval for the population mean, \(\mu\) metres, is calculated as \(5.89 < \mu < 6.75\). Find the unbiased estimates for the population mean and population variance used in this calculation.

9 A farmer grows large amounts of a certain crop. On average, the yield per plant has been 0.75 kg . The farmer has improved the soil in which the crop grows, and she claims that the yield per plant has increased. A random sample of 10 plants grown in the improved soil is chosen. The yields, \(x \mathrm {~kg}\), are summarised as follows. $$\Sigma x = 7.85 \quad \Sigma x ^ { 2 } = 6.19$$

1. Test at the \(5 \%\) significance level whether the farmer's claim is justified, assuming a normal distribution.
2. Find a 95\% confidence interval for the population mean yield for plants grown in the new soil.

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).

9 A random sample of 9 observations of a normally distributed random variable \(X\) gave the following summarised data. $$\Sigma x = 94.5 \quad \Sigma x ^ { 2 } = 993.6$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is 10.2 . Calculate a \(90 \%\) confidence interval for the population mean of \(X\).

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 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).