5.05c Hypothesis test: normal distribution for population mean

In the past, the mean length of a particular variety of worm has been 10.3 cm, with standard deviation 2.6 cm. Following a change in the climate, it is thought that the mean length of this variety of worm has decreased. The lengths of a random sample of 100 worms of this variety are found and the mean of this sample is found to be 9.8 cm. Assuming that the standard deviation remains at 2.6 cm, carry out a test at the 2% significance level of whether the mean length has decreased. [5]

A biologist wishes to test whether the mean concentration \(\mu\), in suitable units, of a certain pollutant in a river is below the permitted level of 0.5. She measures the concentration, \(x\), of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. \(n = 50 \quad \Sigma x = 23.0 \quad \Sigma x^2 = 13.02\)

1. Carry out a test at the 5% significance level of the null hypothesis \(\mu = 0.5\) against the alternative hypothesis \(\mu < 0.5\). [7]

Later, a similar test is carried out at the 5% significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.

1. Given that, in fact, the value of \(\mu\) is 0.4, find the probability of a Type II error. [5]

The lengths, in centimetres, of worms of a certain kind are normally distributed with mean \(\mu\) and standard deviation \(2.3\). An article in a magazine states that the value of \(\mu\) is \(12.7\). A scientist wishes to test whether this value is correct. He measures the lengths, \(x\) cm, of a random sample of \(50\) worms of this kind and finds that \(\sum x = 597.1\). He plans to carry out a test, at the \(1\%\) significance level, of whether the true value of \(\mu\) is different from \(12.7\).

1. State, with a reason, whether he should use a one-tailed or a two-tailed test. [1]
2. Carry out the test. [5]

In the past, the mean annual crop yield from a particular field has been 8.2 tonnes. During the last 16 years, a new fertiliser has been used on the field. The mean yield for these 16 years is 8.7 tonnes. Assume that yields are normally distributed with standard deviation 1.2 tonnes. Carry out a test at the 5\% significance level of whether the mean yield has increased. [5]

From previous years' observations, the lengths of salmon in a river were found to be normally distributed with mean 65 cm. A researcher suspects that pollution in water is restricting growth. To test this theory, she measures the length \(x\) cm of a random sample of \(n\) salmon and calculates that \(\bar{x} = 64.3\) and \(s = 4.9\), where \(s^2\) is the unbiased estimate of the population variance. She then carries out an appropriate hypothesis test.

1. Her test statistic \(z\) has a value of \(-1.807\) correct to 3 decimal places. Calculate the value of \(n\). [3]
2. Using this test statistic, carry out the hypothesis test at the 5% level of significance and state what her conclusion should be. [4]

Records show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km. New driving regulations are introduced and in the next 20 weeks he drives a total of 21 800 km.

1. Stating any assumption(s), test, at the 1% significance level, whether his mean weekly driving distance has decreased. [6]
2. A similar test at the 1% significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context. [2]

The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 20\). He weighs a random sample of 6 bags and the results are summarised as follows: $$\Sigma x = 430 \quad \Sigma x^2 = 40$$ Carry out the test at the 5\% significance level. [7]

Farmer A grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer B grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer A's trees. The masses of apples from Farmer B's trees may be assumed to be normally distributed. A random sample of 10 trees from Farmer B is chosen. The masses, \(x\) kg, of apples produced in a year are summarised as follows. $$\sum x = 152.0 \qquad \sum x^2 = 2313.0$$ Test, at the 5% significance level, whether Farmer B's claim is justified. [6]

As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, \(x\) minutes, was recorded and the results are summarised by $$\sum x = 1570 \quad \text{and} \quad \sum x^2 = 49467.58$$

1. Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. [4]

An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, \(y\) minutes, was recorded and the results are summarised as $$\bar{y} = 30.9 \quad \text{and} \quad s_y^2 = 3.03$$

1. Test, at the 5\% level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly. [7]
2. Explain the relevance of the Central Limit Theorem to the test in part (b). [1]
3. State an assumption you have made in carrying out the test in part (b). [1]

A factory produces steel sheets whose weights \(X\) kg, are such that \(X \sim \text{N}(\mu, \sigma^2)\) A random sample of these sheets is taken and a 95\% confidence interval for \(\mu\) is found to be (29.74, 31.86)

1. Find, to 2 decimal places, the standard error of the mean. [3]
2. Hence, or otherwise, find a 90\% confidence interval for \(\mu\) based on the same sample of sheets. [3]

Using four different random samples, four 90\% confidence intervals for \(\mu\) are to be found.

1. Calculate the probability that at least 3 of these intervals will contain \(\mu\). [3]

A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 80 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.

1. Using a 1\% level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly. [8]
2. State an assumption you made in carrying out the test in part (a). [1]

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]

A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 120 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.

1. Using a 1\% level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly. [8]
2. State an assumption you made in carrying out the test in part (a). [1]

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 computer company repairs large numbers of PCs and wants to estimate the mean time to repair a particular fault. Five repairs are chosen at random from the company's records and the times taken, in seconds, are 205 \quad 310 \quad 405 \quad 195 \quad 320.

1. Calculate unbiased estimates of the mean and the variance of the population of repair times from which this sample has been taken. [4]

It is known from previous results that the standard deviation of the repair time for this fault is 100 seconds. The company manager wants to ensure that there is a probability of at least 0.95 that the estimate of the population mean lies within 20 seconds of its true value.

1. Find the minimum sample size required. [6]

(Total 10 marks)

A biologist investigated whether or not the diet of chickens influenced the amount of cholesterol in their eggs. The cholesterol content of 70 eggs selected at random from chickens fed diet A had a mean value of 198 mg and a standard deviation of 47 mg. A random sample of 90 eggs from chickens fed diet B had a mean cholesterol content of 201 mg and a standard deviation of 23 mg.

1. Stating your hypotheses clearly and using a 5\% level of significance, test whether or not there is a difference between the mean cholesterol content of eggs laid by chickens fed on these two diets. [7]
2. State, in the context of this question, an assumption you have made in carrying out the test in part (a). [2]

The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm. The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm.

1. Test, using a 5\% level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly. [8]
2. State two assumptions you made in carrying out the test in part (a). [2]

A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below. 120.3 \quad 120.1 \quad 120.4 \quad 120.2 \quad 119.9

1. Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company. [5]

The lengths of climbing rope are known to have a standard deviation of 0.2 m. The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of \(n\), lies within 0.05 m of its true value.

1. Find the minimum sample size required. [6]

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)

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]

A sociologist was studying the smoking habits of adults. A random sample of 300 adult smokers from a low income group and an independent random sample of 400 adult smokers from a high income group were asked what their weekly expenditure on tobacco was. The results are summarised below.

1. Using a 5\% significance level, test whether or not the two groups differ in the mean amounts spent on tobacco. [9]
2. Explain briefly the importance of the central limit theorem in this example. [2]

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]