5.05d Confidence intervals: using normal distribution

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 49.7, 50.3, 51.0, 49.5, 49.9 50.1, 50.2, 50.0, 49.6, 49.7.

1. Find unbiased estimates of the mean and variance of the weights of the population of metal containers. [5]

The machine is set to produce metal containers whose weights have a population standard deviation of 0.5 kg.

1. Estimate the limits between which 95\% of the weights of metal containers lie. [4]
2. Determine the 99\% confidence interval for the mean weight of metal containers. [5]

The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm. The heights of the orchids are normally distributed. Given that the population standard deviation is 0.5 cm,

1. estimate limits between which 95\% of the heights of the orchids lie, [3]
2. find a 98\% confidence interval for the mean height of the orchids. [4]

A grower claims that the mean height of this type of orchid is 19.5 cm.

1. Comment on the grower's claim. Give a reason for your answer. [2]

A shop manager wants to find out if customers spend more money when music is playing in the shop. The amount of money spent by a customer in the shop is £\(x\). A random sample of 80 customers, who were shopping without music playing, and an independent random sample of 60 customers, who were shopping with music playing, were surveyed. The results of both samples are summarised in the table below.

	\(\sum x\)	\(\sum x^2\)	Unbiased estimate of mean	Unbiased estimate of variance
Customers shopping without music	5320	392000	\(\bar{x}\)	\(s^2\)
Customers shopping with music	4140	312000	69.0	446.44

1. Find the values of \(\bar{x}\) and \(s^2\). [5]
2. Test, at the 5\% level of significance, whether or not the mean money spent is greater when music is playing in the shop. State your hypotheses clearly. [8]

Roastie's Coffee is sold in packets with a stated weight of 250 g. A supermarket manager claims that the mean weight of the packets is less than the stated weight. She weighs a random sample of 90 packets from their stock and finds that their weights have a mean of 248 g and a standard deviation of 5.4 g.

1. Using a 5\% level of significance, test whether or not the manager's claim is justified. State your hypotheses clearly. [5]
2. Find the 98\% confidence interval for the mean weight of a packet of coffee in the supermarket's stock. [4]
3. State, with a reason, the action you would recommend the manager to take over the weight of a packet of Roastie's Coffee. [2]

Roastie's Coffee company increase the mean weight of their packets to \(\mu\) g and reduce the standard deviation to 3 g. The manager takes a sample of size \(n\) from these new packets. She uses the sample mean \(\bar{X}\) as an estimator of \(\mu\).

1. Find the minimum value of \(n\) such that P\((|\bar{X} - \mu| < 1) \geq 0.98\) [5]

A doctor claims there is a higher mean lung capacity in people who exercise regularly compared to people who do not exercise regularly. He measures the lung capacity, \(x\), of 35 people who exercise regularly and 42 people who do not exercise regularly. His results are summarised in the table below.

1. Test, at the 5\% level of significance, the doctor's claim. State your hypotheses clearly. (6)
2. State any assumptions you have made in testing the doctor's claim. (2) The doctor decides to add another person who exercises regularly to his data. He measures the person's lung capacity and finds \(x = 31.7\)
3. Find the unbiased estimate of the variance for the sample of 36 people who exercise regularly. Give your answer to 3 significant figures. (4)

A restaurant states that its hamburgers contain 20\% fat. Paul claims that the mean fat content of their hamburgers is less than 20\%. Paul takes a random sample of 50 hamburgers from the restaurant and finds that they contain a mean fat content of 19.5\% with a standard deviation of 1.5\% You may assume that the fat content of hamburgers is normally distributed.

1. Find the 90\% confidence interval for the mean fat content of hamburgers from the restaurant. (4)
2. State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers. (2) The restaurant changes the mean fat content of their hamburgers to \(\mu\)\% and adjusts the standard deviation to 2\%. Paul takes a sample of size \(n\) from this new batch of hamburgers. He uses the sample mean \(\bar{X}\) as an estimator of \(\mu\).
3. Find the minimum value of \(n\) such that \(\mathrm{P}(|\bar{X} - \mu| < 0.5) \geq 0.9\) (5)

A random sample \(X_1, X_2, \ldots, X_{10}\) is taken from a normal population with mean 100 and standard deviation 14.

1. Write down the distribution of \(\overline{X}\), the mean of this sample. [2]
2. Find \(\text{Pr}(|\overline{X} - 100| > 5)\). [3]

As part of a research project into the role played by cholesterol in the development of heart disease a random sample of 100 patients was put on a special fish-based diet. A different random sample of 80 patients was kept on a standard high-protein low-fat diet. After several weeks their blood cholesterol was measured and the results summarised in the table below.

Group	Sample size	Mean drop in cholesterol (mg/dl)	Standard deviation
Special diet	100	75	22
Standard diet	80	64	31

1. Stating your hypotheses clearly and using a 5% level of significance, test whether or not the special diet is more effective in reducing blood cholesterol levels than the standard diet. [9]
2. Explain briefly any assumptions you made in order to carry out this test. [2]

As part of her statistics project, Deepa decided to estimate the amount of time A-level students at her school spend on private study each week. She took a random sample of students from those studying Arts subjects, Science subjects and a mixture of Arts and Science subjects. Each student kept a record of the time they spent on private study during the third week of term.

1. Write down the name of the sampling method used by Deepa. [1]
2. Give a reason for using this method and give one advantage this method has over simple random sampling. [2]

The results Deepa obtained are summarised in the table below.

Type of student	Sample size	Mean number of hours
Arts	12	12.6
Science	12	14.1
Mixture	8	10.2

1. Show that an estimate of the mean time spent on private study by A level students at Deepa's school, based on these 32 students is 12.56, to 2 decimal places. [3]

The standard deviation of the time spent on private study by students at the school was 2.48 hours.

1. Assuming that the number of hours spent on private study is normally distributed, find a 95% confidence interval for the mean time spent on private study by A level students at Deepa's school. [4]

A member of staff at the school suggested that A level students should spend on average 12 hours each week on private study.

1. Comment on this suggestion in the light of your interval. [2]

Observations have been made over many years of \(T\), the noon temperature in °C, on 21st 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 \(T\). [5]

Over the years, the standard deviation of \(T\) has been found to be 5.1.

1. 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 °C.

1. Use your interval to comment on the meteorologist's claim. [2]

The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours. Use this model to calculate

1. the probability that a person chosen at random took between 5 and 9 hours to complete the task, [4 marks]
2. the range, symmetrical about the mean, within which 80% of the people's times lie. [5 marks]

It is found that, in fact, 80% of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.

1. Find the standard deviation of the times in the modified model. [3 marks]

The error, \(X\) °C, made in measuring a patient's temperature at a local doctors' surgery may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). The errors, \(x\) °C, made in measuring the temperature of each of a random sample of \(10\) patients are summarised below. $$\sum x = 0.35 \quad \text{and} \quad \sum(x - \bar{x})^2 = 0.12705$$ Construct a \(99\%\) confidence interval for \(\mu\), giving the limits to three decimal places. [5 marks]

A normally distributed variable, \(X\), has unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A sample of 10 values of \(X\) was taken. From these 10 values, a 95% confidence interval for \(\mu\) was calculated to be $$(30.47, 32.93)$$ Use this confidence interval to find unbiased estimates for \(\mu\) and \(\sigma^2\). [4 marks]

Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm. Gerald visited three islands, A, B and C, and measured the length, \(X\) centimetres, of each of a sample of \(n\) sand lizards on each island. The samples may be regarded as random. The data are shown in the table.

Island	\(\sum x\)	\(n\)
A	1384.5	78
B	116.9	7
C	394.6	20

1. Carry out a hypothesis test to investigate whether the data from Island A provide support for Gerald's belief at the 2% significance level. Assume that the standard deviation of the lengths of sand lizards on Island A is 1.8 cm. [7 marks]
2. For Island B, it is also given that $$\sum(x - \bar{x})^2 = 22.64$$
   1. Construct a 95% confidence interval for \(\mu_B\), where \(\mu_B\) centimetres is the mean length of sand lizards on Island B. Assume that the lengths of sand lizards on Island B are normally distributed with unknown standard deviation.
   2. Comment on whether your confidence interval provides support for Gerald's belief.
   [7 marks]
3. Comment on whether the data from Island C provide support for Gerald's belief. [2 marks]

It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years\(^2\).

1. Test at the 5% significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild. [7]
2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years\(^2\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test. [5]
3. Explain whether the Central Limit Theorem is needed in these tests. [1]

In advance of a referendum on independence, the regional assembly of an eastern province of a particular country carried out an opinion poll to assess the strength of the 'Yes' vote. Of the 480 men polled, 264 indicated that they intended to vote 'Yes', and of the 500 women polled, 220 indicated that they intended to vote 'Yes'.

1. Construct an approximate 95\% confidence interval for the difference between the proportion of men who intend to vote 'Yes' and the proportion of women who intend to vote 'Yes'. [6 marks]
2. Comment on a claim that, in the forthcoming referendum, the percentage of men voting 'Yes' will exceed the percentage of women voting 'Yes' by at least 2.5 per cent. Justify your answer. [2 marks]

The continuous random variable \(U\) has a normal distribution with unknown mean \(\mu\) and known variance 1. A random sample of four observations of \(U\) gave the values \(3.9, 2.1, 4.6\) and \(1.4\).

1. Calculate a \(90\%\) confidence interval for \(\mu\). [3]
2. The probability that the sum of four random observations of \(U\) is less than 11 is denoted by \(p\). For each of the end points of the confidence interval in part (i) calculate the corresponding value of \(p\). [5]

A production line has two machines, A and B, for delivering liquid soap into bottles. Each machine is set to deliver a nominal amount of 250 ml, but it is not expected that they will work to a high level of accuracy. In particular, it is known that the ambient temperature affects the rate of flow of the liquid and leads to variation in the amounts delivered. The operators think that machine B tends to deliver a somewhat greater amount than machine A, no matter what the ambient temperature. This is being investigated by an experiment. A random sample of 10 results from the experiment is shown below. Each column of data is for a different ambient temperature.

Ambient temperature	\(T_1\)	\(T_2\)	\(T_3\)	\(T_4\)	\(T_5\)	\(T_6\)	\(T_7\)	\(T_8\)	\(T_9\)	\(T_{10}\)
Amount delivered by machine A	246.2	251.6	252.0	246.6	258.4	251.0	247.5	247.1	248.1	253.4
Amount delivered by machine B	248.3	252.6	252.8	247.2	258.8	250.0	247.2	247.9	249.0	254.5

1. Use an appropriate \(t\) test to examine, at the 5\% level of significance, whether the mean amount delivered by machine B may be taken as being greater than that delivered by machine A, stating carefully your null and alternative hypotheses and the required distributional assumption. [11]
2. Using the data for machine A in the table above, provide a two-sided 95\% confidence interval for the mean amount delivered by this machine, stating the required distributional assumption. Explain whether you would conclude that the machine appears to be working correctly in terms of the nominal amount as set. [7]

An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.

1. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms. [3]
2. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms. [3]
3. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms. [3]
4. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor. [5]
5. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a 95\% confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified. [4]

William Sealy, a biochemistry student, is doing work experience at a brewery. One of his tasks is to monitor the specific gravity of the brewing mixture during the brewing process. For one particular recipe, an initial specific gravity of 1.040 is required. A random sample of 9 measurements of the specific gravity at the start of the process gave the following results. 1.046 \quad 1.048 \quad 1.039 \quad 1.055 \quad 1.038 \quad 1.054 \quad 1.038 \quad 1.051 \quad 1.038

1. William has to test whether the specific gravity of the mixture meets the requirement. Why might a \(t\) test be used for these data and what assumption must be made? [3]
2. Carry out the test using a significance level of 10\%. [9]
3. Find a 95\% confidence interval for the true mean specific gravity of the mixture and explain what is meant by a 95\% confidence interval. [6]

A film-buff is interested in how long it takes for the credits to roll at the end of a movie. She takes a random sample of 20 movies from those that she has bought on DVD and finds that the credits on these films last for a total of 46 minutes and 15 seconds

1. Assuming that the time for the credits to roll follows a Normal distribution with a standard deviation of 23 seconds, use her data to calculate a 90% confidence interval for the mean time taken for the credits to roll. [5]
2. Find the minimum number of movies she would need to have included in her sample for her confidence interval to have a width of less than 10 seconds. [5]
3. Explain why her sample might not be representative of all movies. [1]

A teacher gives each student in his class a list of 30 numbers. All the numbers have been generated at random by a computer from a normal distribution with a fixed mean and variance. The teacher tells the class that the variance of the distribution is 25 and asks each of them to calculate a 95\% confidence interval based on their list of numbers. The sum of the numbers given to one student is 1419.

1. Find the confidence interval that should be obtained by this student. [5]

Assuming that all the students calculate their confidence intervals correctly,

1. state the proportion of the students you would expect to have a confidence interval that includes the true mean of the distribution, [1]
2. explain why the probability of any one student's confidence interval including the true mean is not 0.95 [1]

A random sample of 15 tomatoes is taken and the weight \(x\) grams of each tomato is found. The results are summarised by \(\sum x = 208\) and \(\sum x^2 = 2962\).

1. Assuming that the weights of the tomatoes are normally distributed, calculate the 90\% confidence interval for the variance \(\sigma^2\) of the weights of the tomatoes. [7]
2. State with a reason whether or not the confidence interval supports the assertion \(\sigma^2 = 3\). [2]