5.05d Confidence intervals: using normal distribution

Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml, are given in the table. \includegraphics{figure_7} One statistician suggests performing a two-sample \(t\)-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.

1. Stating your hypotheses clearly and using a 5\% significance level, carry out this test. (You may assume \(\bar{x}_A = 26.125\), \(s_A^2 = 7.84\), \(\bar{x}_B = 25\), \(s_B^2 = 4\) and \(\sigma_A^2 = \sigma_B^2\).) [7]

Another statistician suggests analysing these data using a paired \(t\)-test.

1. [(b)] Using a 5\% significance level, carry out this test. [9]
2. State which of these two tests you consider to be more appropriate. Give a reason for your choice. [1]

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]

Brickland and Goodbrick are two manufacturers of bricks. The lengths of the bricks produced by each manufacturer can be assumed to be normally distributed. A random sample of 20 bricks is taken from Brickland and the length, \(x\) mm, of each brick is recorded. The mean of this sample is 207.1 mm and the variance is 3.2 mm².

1. Calculate the 98\% confidence interval for the mean length of brick from Brickland. [4]

A random sample of 10 bricks is selected from those manufactured by Goodbrick. The length of each brick, \(y\) mm, is recorded. The results are summarised as follows. \(\sum y = 2046.2\) \(\sum y^2 = 418785.4\) The variances of the length of brick for each manufacturer are assumed to be the same.

1. [(b)] Find a 90\% confidence interval for the value by which the mean length of brick made by Brickland exceeds the mean length of brick made by Goodbrick. [8]

(Total 12 marks)

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]

Two machines \(A\) and \(B\) produce the same type of component in a factory. The factory manager wishes to know whether the lengths, \(x\) cm, of the components produced by the two machines have the same mean. The manager took a random sample of components from each machine and the results are summarised in the table below. \includegraphics{figure_4} The lengths of components produced by the machines can be assumed to follow normal distributions.

1. Use a two tail test to show, at the 10\% significance level, that the variances of the lengths of components produced by each machine can be assumed to be equal. [4]
2. Showing your working clearly, find a 95\% confidence interval for \(\mu_A - \mu_B\), where \(\mu_A\) and \(\mu_B\) are the mean lengths of the populations of components produced by machine \(A\) and machine \(B\) respectively. [7]

There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm.

1. [(c)] State, giving your reason, whether or not the factory manager should be concerned. [2]

The length \(X\) mm of a spring made by a machine is normally distributed N(\(\mu, \sigma^2\)). A random sample of 20 springs is selected and their lengths measured in mm. Using this sample the unbiased estimates of \(\mu\) and \(\sigma^2\) are \(\bar{x} = 100.6\), \(s^2 = 1.5\). Stating your hypotheses clearly test, at the 10\% level of significance,

1. whether or not the variance of the lengths of springs is different from 0.9, [6]
2. whether or not the mean length of the springs is greater than 100 mm. [6]

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]

A machine is filling bottles of milk. A random sample of 16 bottles was taken and the volume of milk in each bottle was measured and recorded. The volume of milk in a bottle is normally distributed and the unbiased estimate of the variance, \(s^2\), of the volume of milk in a bottle is 0.003

1. Find a 95\% confidence interval for the variance of the population of volumes of milk from which the sample was taken. [5] The machine should fill bottles so that the standard deviation of the volumes is equal to 0.07
2. Comment on this with reference to your 95\% confidence interval. [3]

A farmer set up a trial to assess whether adding water to dry feed increases the milk yield of his cows. He randomly selected 22 cows. Thirteen of the cows were given dry feed and the other 9 cows were given the feed with water added. The milk yields, in litres per day, were recorded with the following results.

	Sample size	Mean	\(s^2\)
Dry feed	13	25.54	2.45
Feed with water added	9	27.94	1.02

You may assume that the milk yield from cows given the dry feed and the milk yield from cows given the feed with water added are from independent normal distributions.

1. Test, at the 10\% level of significance, whether or not the variances of the populations from which the samples are drawn are the same. State your hypotheses clearly. [5]
2. Calculate a 95\% confidence interval for the difference between the two mean milk yields. [7]
3. Explain the importance of the test in part (a) to the calculation in part (b). [2]

A machine fills jars with jam. The weight of jam in each jar is normally distributed. To check the machine is working properly the contents of a random sample of 15 jars are weighed in grams. Unbiased estimates of the mean and variance are obtained as $$\mu = 560 \quad s^2 = 25.2$$ Calculate a 95\% confidence interval for,

1. the mean weight of jam, [4]
2. the variance of the weight of jam. [5]

A weight of more than 565g is regarded as too high and suggests the machine is not working properly.

1. Use appropriate confidence limits from parts (a) and (b) to find the highest estimate of the proportion of jars that weigh too much. [5]

A teacher wishes to test whether playing background music enables students to complete a task more quickly. The same task was completed by 15 students, divided at random into two groups. The first group had background music playing during the task and the second group had no background music playing. The times taken, in minutes, to complete the task are summarised below.

	Sample size \(n\)	Standard deviation \(s\)	Mean \(\bar{x}\)
With background music	8	4.1	15.9
Without background music	7	5.2	17.9

You may assume that the times taken to complete the task by the students are two independent random samples from normal distributions.

1. Stating your hypotheses clearly, test, at the 10\% level of significance, whether or not the variances of the times taken to complete the task with and without background music are equal. [5]
2. Find a 99\% confidence interval for the difference in the mean times taken to complete the task with and without background music. [7]

Experiments like this are often performed using the same people in each group.

1. Explain why this would not be appropriate in this case. [1]

A random sample of 15 strawberries is taken from a large field and the weight \(x\) grams of each strawberry is recorded. The results are summarised below. $$\sum x = 291 \quad \sum x^2 = 5968$$ Assume that the weights of strawberries are normally distributed. Calculate a 95\% confidence interval for

1. (i) the mean of the weights of the strawberries in the field, (ii) the variance of the weights of the strawberries in the field. [12]

Strawberries weighing more than 23g are considered to be less tasty.

1. Use appropriate confidence limits from part (a) to find the highest estimate of the proportion of strawberries that are considered to be less tasty. [4]

Two independent random samples \(X_1, X_2, ..., X_n\) and \(Y_1, Y_2, Y_3, Y_4\) were taken from different normal populations with a common standard deviation \(\sigma\). The following sample statistics were calculated. $$s_x = 14.67 \quad s_y = 12.07$$ Find the 99\% confidence interval for \(\sigma^2\) based on these two samples. [5]

The weights of the contents of breakfast cereal boxes are normally distributed. A manufacturer changes the style of the boxes but claims that the weight of the contents remains the same. A random sample of 6 old style boxes had contents with the following weights (in grams). 512, 503, 514, 506, 509, 515 The weights, \(y\) grams, of the contents of an independent random sample of 5 new style boxes gave $$\bar{y} = 504.8 \text{ and } s_y = 3.420$$

1. Use a two-tail test to show, at the 10\% level of significance, that the variances of the weights of the contents of the old and new style boxes can be assumed to be equal. State your hypotheses clearly. [5]
2. Showing your working clearly, find a 90\% confidence interval for \(\mu_x - \mu_y\) where \(\mu_x\) and \(\mu_y\) are the mean weights of the contents of old and new style boxes respectively. [7]
3. With reference to your confidence interval comment on the manufacturer's claim. [2]

A nutritionist studied the levels of cholesterol, \(X\) mg/cm³, of male students at a large college. She assumed that \(X\) was distributed \(\text{N}(\mu, \sigma^2)\) and examined a random sample of 25 male students. Using this sample she obtained unbiased estimates of \(\mu\) and \(\sigma^2\) as $$\hat{\mu} = 1.68, \quad \hat{\sigma}^2 = 1.79.$$

1. Find a 95% confidence interval for \(\mu\). [4]
2. Obtain a 95% confidence interval for \(\sigma^2\). [5]

A cholesterol reading of more than 2.5 mg/cm³ is regarded as high.

1. Use appropriate confidence limits from parts \((a)\) and \((b)\) to find the lowest estimate of the proportion of male students in the college with high cholesterol. [4]

A newspaper runs a daily Sudoku. A random sample of 10 people took the following times, in minutes, to complete the Sudoku. 5.0 \quad 4.5 \quad 4.7 \quad 5.3 \quad 5.2 \quad 4.1 \quad 5.3 \quad 4.8 \quad 5.5 \quad 4.6 Given that the times to complete the Sudoku follow a normal distribution,

1. calculate a 95\% confidence interval for
   1. the mean,
   2. the variance,
   of the times taken by people to complete the Sudoku. [13] The newspaper requires the average time needed to complete the Sudoku to be 5 minutes with a standard deviation of 0.7 minutes.
2. Comment on whether or not the Sudoku meets this requirement. Give a reason for your answer. [3]

Murni is investigating the annual salary of people from a particular town. She takes a random sample of 200 people from the town and records their annual salary. The mean annual salary is £28 500 and the standard deviation is £5100 Calculate a 97% confidence interval for the population mean of annual salaries for the people who live in the town, giving your values to the nearest pound. [3 marks]

The continuous random variable \(X\) has the distribution \(\text{N}(\mu, 30)\). 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. [4]

A coffee shop produces biscuits to sell. The masses, in grams, of the biscuits follow a normal distribution with mean \(\mu\). Eight biscuits are chosen at random and their masses, in grams, are recorded. The results are given below. 32.1 \quad 29.9 \quad 31.0 \quad 31.1 \quad 32.5 \quad 30.8 \quad 30.7 \quad 31.5

1. Calculate a 95\% confidence interval for \(\mu\) based on this sample. [7]
2. Explain the relevance or otherwise of the Central Limit Theorem in your calculations. [1]

Rugby players sometimes use protein powder to aid muscle increase. The monthly weight gains of rugby players taking protein powder may be modelled by a normal distribution having a standard deviation of 40 g and a mean which may depend on the type of protein powder they consume. A rugby team coach gives the same amount of protein powder over a trial month to 22 randomly selected players. Protein powder \(A\) was used by 12 players, randomly selected, and their mean weight gain was 900 g. Protein powder \(B\) was used by the other 10 players and their mean weight gain was 870 g. Let \(\mu_A\) and \(\mu_B\) be the mean monthly weight gains, in grams, of the populations of rugby players who use protein powder \(A\) and protein powder \(B\) respectively.

1. Calculate a 98\% confidence interval for \(\mu_A - \mu_B\). [4]
2. In the given context, what can you conclude from your answer to part (a)? Give a reason for your answer. [2]
3. Find the confidence level of the largest confidence interval that would lead the coach to favour protein powder \(A\) over protein powder \(B\). [4]
4. State one non-statistical assumption you have made in order to reach these conclusions. [1]

During practice sessions, a basketball coach makes his players run several 'line drills'.

1. He records the times taken, in seconds, by one of his players to run the first 'line drill' on a random sample of 8 practice sessions. The results are shown below. 29.4 \quad 31.1 \quad 28.9 \quad 30.0 \quad 29.9 \quad 30.4 \quad 29.7 \quad 30.2 Assuming that these data come from a normal distribution with mean \(\mu\) and variance 0.6, calculate a 95\% confidence interval for \(\mu\). [5]
2. State the two ways in which the method used to calculate the confidence interval in part (a) would change if the variance were unknown. [2]
3. During a practice session, a player recorded a mean time of 35.6 seconds for 'line drills'.
   1. Give a reason why this player may not be the same as the player in part (a).
   2. Give a reason why this player could be the same as the player in part (a). [2]

A factory manufactures a certain type of string. In order to ensure the quality of the product, a random sample of 10 pieces of string is taken every morning and the breaking strength of each piece, in Newtons, is measured. One morning, the results are as follows. $$68.1 \quad 70.4 \quad 68.6 \quad 67.7 \quad 71.3 \quad 67.6 \quad 68.9 \quad 70.2 \quad 68.4 \quad 69.8$$ You may assume that this is a random sample from a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma^2\).

1. Determine a 95% confidence interval for \(\mu\). [9]
2. The factory manager is given these results and he asks 'Can I assume that the confidence interval that you have given me contains \(\mu\) with probability 0.95?' Explain why the answer to this question is no and give a correct interpretation. [2]

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.

1. Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place. [2 marks]
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. [1 mark]
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. [1 mark]

The mean of a random sample of \(n\) observations drawn from a normal distribution with mean \(\mu\) and variance \(\sigma^2\) is denoted by \(\bar{X}\). It is given that P(\(\mu - 0.5\sigma < \bar{X} < \mu + 0.5\sigma\)) > 0.95.

1. Find the smallest possible value of \(n\). [5]
2. With this value of \(n\), find P(\(\bar{X} > \mu - 0.1\sigma\)). [3]