CI from raw data list

3 A consumer group, interested in the mean fat content of a particular type of sausage, takes a random sample of 20 sausages and sends them away to be analysed. The percentage of fat in each sausage is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 26 & 27 & 28 & 28 & 28 & 29 & 29 & 30 & 30 & 31 & 32 & 32 & 32 & 33 & 33 & 34 & 34 & 34 & 35 & 35 \end{array}$$ Assume that the percentage of fat is normally distributed with mean \(\mu\), and that the standard deviation is known to be 3 .

1. Calculate a 98\% confidence interval for the population mean percentage of fat.
2. The manufacturer claims that the mean percentage of fat in sausages of this type is 30 . Use your answer to part (i) to determine whether the consumer group should accept this claim.

3 Each of a random sample of 15 students was asked how long they spent revising for an exam. The results, in minutes, were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 50 & 70 & 80 & 60 & 65 & 110 & 10 & 70 & 75 & 60 & 65 & 45 & 50 & 70 & 50 \end{array}$$ Assume that the times for all students are normally distributed with mean \(\mu\) minutes and standard deviation 12 minutes.

1. Calculate a \(92 \%\) confidence interval for \(\mu\).
2. Explain what is meant by a \(92 \%\) confidence interval for \(\mu\).
3. Explain what is meant by saying that a sample is 'random'.

5 The amount of money, in dollars, spent by a customer on one visit to a certain shop is modelled by the distribution \(\mathrm { N } ( \mu , 1.94 )\). In the past, the value of \(\mu\) has been found to be 20.00 , but following a rearrangement in the shop, the manager suspects that the value of \(\mu\) has changed. He takes a random sample of 6 customers and notes how much they each spend, in dollars. The results are as follows.
15.50
17.60
17.30
22.00
23.50
31.00 The manager carries out a hypothesis test using a significance level of \(\alpha \%\). The test does not support his suspicion. Find the largest possible value of \(\alpha\).

3 The times, in minutes, taken by competitors to complete a puzzle have mean \(\mu\) and standard deviation 3 . The times taken by a random sample of 10 competitors are noted and the results are given below. \(\begin{array} { l l l } 25.2 & 26.8 & 18.5 \end{array}\) 25.5
30.1 \(28.9 \quad 27.0\) \(26.1 \quad 26.0\) 24.9

1. Stating a necessary assumption, calculate a \(97 \%\) confidence interval for \(\mu\).
2. Two more random samples, each of 10 competitors, are taken. Their times are used to calculate two more \(97 \%\) confidence intervals for \(\mu\). Find the probability that neither of these intervals contains the true value of \(\mu\).

8 In a crossword competition the times, \(x\) minutes, taken by a random sample of 6 entrants to complete a crossword are summarised as follows. $$\Sigma x = 210.9 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 151.2$$ The time to complete a crossword has a normal distribution with mean \(\mu\) minutes. Calculate a \(95 \%\) confidence interval for \(\mu\). Assume now that the standard deviation of the population is known to be 5.6 minutes. Find the smallest sample size that would lead to a \(95 \%\) confidence interval for \(\mu\) of width at most 5 minutes.

The time taken for a randomly chosen student at College \(P\) to complete a particular puzzle has a normal distribution with mean \(\mu\) minutes. The times, \(x\) minutes, are recorded for a random sample of 8 students chosen from the college. The results are summarised as follows. $$\Sigma x = 42.8 \quad \Sigma x ^ { 2 } = 236.0$$ Find a 95\% confidence interval for \(\mu\). A test is carried out on this sample data, at the \(10 \%\) significance level. The test supports the claim that \(\mu > k\). Find the greatest possible value of \(k\). A random sample, of size 12, is taken from the students at College \(Q\). Their times to complete the puzzle give a sample mean of 4.60 minutes and an unbiased variance estimate of 1.962 minutes \({ } ^ { 2 }\). Use a 2 -sample test at the \(10 \%\) significance level to test whether the mean time for students at College \(Q\) to complete the puzzle is less than the mean time for students at College \(P\) to complete the puzzle. You should state any assumptions necessary for the test to be valid.

3. The drying times of paint can be assumed to be normally distributed. A paint manufacturer paints 10 test areas with a new paint. The following drying times, to the nearest minute, were recorded. $$82 , \quad 98 , \quad 140 , \quad 110 , \quad 90 , \quad 125 , \quad 150 , \quad 130 , \quad 70 , \quad 110 .$$

1. Calculate unbiased estimates for the mean and the variance of the population of drying times of this paint. Given that the population standard deviation is 25 ,
2. find a 95\% confidence interval for the mean drying time of this paint. Fifteen similar sets of tests are done and the \(95 \%\) confidence interval is determined for each set.
3. Estimate the expected number of these 15 intervals that will enclose the true value of the population mean \(\mu\).

7. A petrol pump is tested regularly to check that the reading on its gauge is accurate. The random variable \(X\), in litres, is the quantity of petrol actually dispensed when the gauge reads 10.00 litres. \(X\) is known to have distribution \(X \sim \mathrm {~N} \left( \mu , 0.08 ^ { 2 } \right)\)

1. Eight random tests gave the following values of \(x\) $$\begin{array} { l l l l l l l l } 10.01 & 9.97 & 9.93 & 9.99 & 9.90 & 9.95 & 10.13 & 9.94 \end{array}$$
   1. Find a 95\% confidence interval for \(\mu\) to 2 decimal places.
   2. Use your result to comment on the accuracy of the petrol gauge.
2. A sample mean of 9.96 litres was obtained from a random sample of \(n\) tests. A \(90 \%\) confidence interval for \(\mu\) gave an upper limit of less than 10.00 litres. Find the minimum value of \(n\).

3 The height, in metres, of adult male African elephants may be assumed to be normally distributed with mean \(\mu\) and standard deviation 0.20 . The heights of a sample of 12 such elephants were measured with the following results, in metres. $$\begin{array} { l l l l l l l l l l l l } 3.37 & 3.45 & 2.93 & 3.42 & 3.49 & 3.67 & 2.96 & 3.57 & 3.36 & 2.89 & 3.22 & 2.91 \end{array}$$

1. Stating a necessary assumption, construct a \(98 \%\) confidence interval for \(\mu\). (6 marks)
2. The mean height of adult male Asian elephants is known to be 2.90 metres. Using your confidence interval, state, with a reason, what can be concluded about the mean heights of adult males in these two types of elephant.

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]

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]

1. A museum is open to the public for six hours a day from Monday to Friday every week. The number of visitors, \(V\), to the museum on ten randomly chosen days were as follows:

$$\begin{array} { l l l l l l l l l l } 182 & 172 & 113 & 99 & 168 & 183 & 135 & 129 & 150 & 108 \end{array}$$

1. Calculate an unbiased estimate of the mean of \(V\). Assuming that \(V\) is normally distributed with a variance of 130 ,
2. find a 95\% confidence interval for the mean of \(V\).

4. A town council is concerned that the mean price of renting two bedroom flats in the town has exceeded \(\pounds 650\) per month. A random sample of eight two bedroom flats gave the following results, \(\pounds x\), per month. $$705 , \quad 640 , \quad 560 , \quad 680 , \quad 800 , \quad 620 , \quad 580 , \quad 760$$ [You may assume \(\sum x = 5345 \quad \sum x ^ { 2 } = 3621025\) ]

1. Find a 90\% confidence interval for the mean price of renting a two bedroom flat.
2. State an assumption that is required for the validity of your interval in part (a).
3. Comment on whether or not the town council is justified in being concerned. Give a reason for your answer.

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.

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 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 (a) and (b).

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. 10.03 \quad 10.02 \quad 9.98 \quad 10.06 \quad 10.08 \quad 10.01 Calculate a 99\% confidence interval for the mean diameter of all discs produced by the machine. [5] Deduce a 99\% confidence interval for the mean circumference of all discs produced by the machine. [1]

The heights, in metres, of a random sample of 8 trees of a particular type are as follows. 14.2 11.3 10.8 8.4 12.8 11.5 12.1 9.2 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. [6]

The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. $$498 \quad 502 \quad 500 \quad 496 \quad 509 \quad 504 \quad 511 \quad 497 \quad 506 \quad 499$$

1. Find unbiased estimates of the mean and the variance of the population from which this sample was taken. [5]

Given that the population standard deviation is 5.0 g,

1. estimate limits, to 2 decimal places, between which 90\% of the weights of the tubs lie, [2]
2. find a 95\% confidence interval for the mean weight of the tubs. [5]

A second random sample of 15 tubs was found to have a mean weight of 501.9 g.

1. Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the mean weight of these tubs is greater than 500 g. [5]

The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. 498 502 500 496 509 504 511 497 506 499

1. Find unbiased estimates of the mean and the variance of the population from which this sample was taken. [5]

Given that the population standard deviation is 5.0 g,

1. estimate limits, to 2 decimal places, between which 90\% of the weights of the tubs lie, [2]
2. find a 95\% confidence interval for the mean weight of the tubs. [5]

A second random sample of 15 tubs was found to have a mean weight of 501.9 g.

1. Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the mean weight of these tubs is greater than 500 g. [5]

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]

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]

A supervisor wishes to check the typing speed of a new typist. On 10 randomly selected occasions, the supervisor records the time taken for the new typist to type 100 words. The results, in seconds, are given below. 110, 125, 130, 126, 128, 127, 118, 120, 122, 125 The supervisor assumes that the time taken to type 100 words is normally distributed.

1. Calculate a 95\% confidence interval for
   1. the mean,
   2. the variance
   of the population of times taken by this typist to type 100 words. [13]

The supervisor requires the average time needed to type 100 words to be no more than 130 seconds and the standard deviation to be no more than 4 seconds.

1. [(b)] Comment on whether or not the supervisor should be concerned about the speed of the new typist. [3]

The weights, in grams, of apples are assumed to follow a normal distribution. The weights of apples sold by a supermarket have variance \(\sigma_1^2\). A random sample of 4 apples from the supermarket had weights 114, 100, 119, 123.

1. Find a 95\% confidence interval for \(\sigma_1^2\). [7]

The weights of apples sold on a market stall have variance \(\sigma_M^2\). A second random sample of 7 apples was taken from the market stall. The sample variance \(s_M^2\) of the apples was 318.8.

1. [(b)] Stating your hypotheses clearly test, at the 1\% level of significance, whether or not there is evidence that \(\sigma_M^2 > \sigma_1^2\). [5]

A town council is concerned that the mean price of renting two bedroom flats in the town has exceeded £650 per month. A random sample of eight two bedroom flats gave the following results, £\(x\), per month. 705, 640, 560, 680, 800, 620, 580, 760 [You may assume \(\sum x = 5345\) and \(\sum x^2 = 3621025\)]

1. Find a 90\% confidence interval for the mean price of renting a two bedroom flat. [6]
2. State an assumption that is required for the validity of your interval in part (a). [1]
3. Comment on whether or not the town council is justified in being concerned. Give a reason for your answer. [2]