5.05a Sample mean distribution: central limit theorem

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]

A sample of size 5 is taken from a population that is normally distributed with mean 10 and standard deviation 3. Find the probability that the sample mean lies between 7 and 10. (Total 6 marks)

A report on the health and nutrition of a population stated that the mean height of three-year old children is 90 cm and the standard deviation is 5 cm. A sample of 100 three-year old children was chosen from the population.

1. Write down the approximate distribution of the sample mean height. Give a reason for your answer. [3]
2. Hence find the probability that the sample mean height is at least 91 cm. [3]

A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57. [5]

Explain what you understand by the Central Limit Theorem. [3]

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]

The weights of pears, \(P\) grams, are normally distributed with a mean of 110 and a standard deviation of 8. Geoff buys a bag of 16 pears.

1. Write down the distribution of \(\overline{P}\), the mean weight of the 16 pears. [2]
2. Find P\((110 < \overline{P} < 113)\). [3]

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]

A secretarial agency carefully assesses the work of a new recruit, with the following results after 150 pages:

No of errors	0	1	2	3	4	5	6
No of pages	16	38	41	29	17	7	2

1. Find the mean and variance of the number of errors per page. [4 marks]
2. Explain how these results support the idea that the number of errors per page follows a Poisson distribution. [1 mark]
3. After two weeks at the agency, the secretary types a fresh piece of work, six pages long, which is found to contain 15 errors. The director suspects that the secretary was trying especially hard during the early period and that she is now less conscientious. Using a Poisson distribution with the mean found in part (a), test this hypothesis at the 5% significance level. [5 marks]

A certain Sixth Former is late for school once a week, on average. In a particular week of 5 days, find the probability that

1. he is not late at all, [2 marks]
2. he is late more than twice. [3 marks]

In a half term of seven weeks, lateness on more than ten occasions results in loss of privileges the following half term.

1. Use the Normal approximation to estimate the probability that he loses his privileges. [7 marks]

A certain type of steel is produced in a foundry. It has flaws (small bubbles) randomly distributed, and these can be detected by X-ray analysis. On average, there are 0·1 bubbles per cm³, and the number of bubbles per cm³ has a Poisson distribution. In an ingot of 40 cm³, find

1. the probability that there are less than two bubbles, [3 marks]
2. the probability that there are more than 3 but less than 10 bubbles. [3 marks]

A new machine is being considered. Its manufacturer claims that it produces fewer bubbles per cm³. In a sample ingot of 60 cm³, there is just one bubble.

1. Carry out a hypothesis test at the 1% significance level to decide whether the new machine is better. State your hypotheses and conclusion carefully. [6 marks]

It is thought that a random variable \(X\) has a Poisson distribution whose mean, \(\lambda\), is equal to 8. Find the critical region to test the hypothesis \(H_0 : \lambda = 8\) against the hypothesis \(H_1 : \lambda < 8\), working at the 1\% significance level. [5 marks]

A centre for receiving calls for the emergency services gets an average of 3.5 emergency calls every minute. Assuming that the number of calls per minute follows a Poisson distribution,

1. find the probability that more than 6 calls arrive in any particular minute. [3 marks] Each operator takes a mean time of 2 minutes to deal with each call, and therefore seven operators are necessary to cope with the average demand.
2. Find how many operators are required for there to be a 99\% probability that a call can be dealt with immediately. [3 marks] It is found from experience that a major disaster creates a surge of emergency calls. Taking the null hypothesis \(H_0\) that there is no disaster,
3. find the number of calls that need to be received in one minute to disprove \(H_0\) at the 0.1 \% significance level. [3 marks]

The continuous random variable \(T\) is equally likely to take any value from 5.0 to 11.0 inclusive.

1. Sketch the graph of the probability density function of \(T\). [2]
2. Write down the value of E(\(T\)) and find by integration the value of Var(\(T\)). [5]
3. A random sample of 48 observations of \(T\) is obtained. Find the approximate probability that the mean of the sample is greater than 8.3, and explain why the answer is an approximation. [6]

The discrete random variable \(H\) takes values 1, 2, 3 and 4. It is given that E(\(H\)) = 2.5 and Var(\(H\)) = 1.25. The mean of a random sample of 50 observations of \(H\) is denoted by \(\bar{H}\). Use a suitable approximation to find P(\(\bar{H} < 2.6\)). [5]

The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).

1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [2]
2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate P\((4 \leq X < 8)\). [3]
3. At a busy time of the day, \(\lambda = 30\).
   1. Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
   2. Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]

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]

A random variable \(X\) has an exponential distribution with probability density function \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\), where \(\lambda\) is a positive constant.

1. Verify that \(\int_0^{\infty} f(x) \, dx = 1\) and sketch \(f(x)\). [5]
2. In this part of the question you may use the following result. $$\int_0^{\infty} x^r e^{-\lambda x} \, dx = \frac{r!}{\lambda^{r+1}} \text{ for } r = 0, 1, 2, \ldots$$ Derive the mean and variance of \(X\) in terms of \(\lambda\). [6]

The random variable \(X\) is used to model the lifetime, in years, of a particular type of domestic appliance. The manufacturer of the appliance states that, based on past experience, the mean lifetime is 6 years.

1. Let \(\overline{X}\) denote the mean lifetime, in years, of a random sample of 50 appliances. Write down an approximate distribution for \(\overline{X}\). [4]
2. A random sample of 50 appliances is found to have a mean lifetime of 7.8 years. Does this cast any doubt on the model? [3]

A company produces kettlebells whose weights are normally distributed with mean \(16\) kg and standard deviation \(0.08\) kg.

1. Find the probability that the weight of a randomly selected kettlebell is greater than \(16.05\) kg. [2]

The company trials a new production method. It needs to check that the mean is still \(16\) kg. It assumes that the standard deviation is unchanged. The company takes a random sample of 25 kettlebells and it decides to reject the new production method if the sample mean does not round to \(16\) kg to the nearest \(100\) g.

1. Find the probability that the new production method will be rejected if, in fact, the mean is still \(16\) kg. [4]

The company decides instead to use a 5\% significance test. A random sample of 25 kettlebells is selected and the mean is found to be \(16.02\) kg.

1. Carry out the test to determine whether or not the new production method will be rejected. [6]

The greatest weight \(W\) N that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80. A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N.

1. Test at the 1% significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support. [7]
2. State an assumption needed in carrying out the test in part (a). [1]
3. Explain whether it is necessary to use the central limit theorem in carrying out the test. [1]

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]