5.05a Sample mean distribution: central limit theorem

6 The time, in minutes, for Anjan's journey to work on Mondays has mean 38.4 and standard deviation 6.9.

1. Find the probability that Anjan's mean journey time for a random sample of 30 Mondays is between 38 and 40 minutes.
   Anjan wishes to test whether his mean journey time is different on Tuesdays. He chooses a random sample of 30 Tuesdays and finds that his mean journey time for these 30 Tuesdays is 40.2 minutes. Assume that the standard deviation for his journey time on Tuesdays is 6.9 minutes.
2. 1. State, with a reason, whether Anjan should use a one-tail or a two-tail test.
   2. Carry out the test at the \(10 \%\) significance level.
   3. Explain whether it was necessary to use the Central Limit theorem in part (b)(ii).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The number of words in History essays by students at a certain college has mean \(\mu\) and standard deviation 1420.

1. The mean number of words in a random sample of 125 History essays was found to be 4820 . Calculate a \(98 \%\) confidence interval for \(\mu\).
2. Another random sample of \(n\) History essays was taken. Using this sample, a \(95 \%\) confidence interval for \(\mu\) was found to be 4700 to 4980 , both correct to the nearest integer. Find the value of \(n\).

1 The standard deviation of the heights of adult males is 7.2 cm . The mean height of a sample of 200 adult males is found to be 176 cm .

1. Calculate a \(97.5 \%\) confidence interval for the mean height of adult males.
2. State a necessary condition for the calculation in part (i) to be valid.

3 A population has mean 12 and standard deviation 2.5. A large random sample of size \(n\) is chosen from this population and the sample mean is denoted by \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } < 12.2 ) = 0.975\), correct to 3 significant figures, find the value of \(n\).

2 The standard deviation of the volume of drink in cans of Koola is 4.8 centilitres. A random sample of 180 cans is taken and the mean volume of drink in these 180 cans is found to be 330.1 centilitres.

1. Calculate a \(95 \%\) confidence interval for the mean volume of drink in all cans of Koola. Give the end-points of your interval correct to 1 decimal place.
2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).

7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively.

1. Find \(\mathrm { P } ( X + Y = 3 )\).
2. Given that \(X + Y = 3\), find \(\mathrm { P } ( X = 2 )\).
3. A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is more than 2.2.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The heights of a certain species of animal have been found to have mean 65.2 cm and standard deviation 7.1 cm . A researcher suspects that animals of this species in a certain region are shorter on average than elsewhere. She takes a large random sample of \(n\) animals of this species from this region and finds that their mean height is 63.2 cm . She then carries out an appropriate hypothesis test.

1. She finds that the value of the test statistic \(z\) is - 2.182 , correct to 3 decimal places.
   1. Stating a necessary assumption, calculate the value of \(n\).
   2. Carry out the hypothesis test at the \(4 \%\) significance level.
   3. Explain why it was necessary to use the Central Limit theorem in carrying out the test.

2 The mean and standard deviation of the time spent by people in a certain library are 29 minutes and 6 minutes respectively.

1. Find the probability that the mean time spent in the library by a random sample of 120 people is more than 30 minutes.
2. Explain whether it was necessary to assume that the time spent by people in the library is normally distributed in the solution to part (i).

4 The lifetimes, \(X\) hours, of a random sample of 50 batteries of a certain kind were found. The results are summarised by \(\Sigma x = 420\) and \(\Sigma x ^ { 2 } = 27530\).

1. Calculate an unbiased estimate of the population mean of \(X\) and show that an unbiased estimate of the population variance is 490 , correct to 3 significant figures.
2. The lifetimes of a further large sample of \(n\) batteries of this kind were noted, and the sample mean, \(\bar { X }\), was found. Use your estimates from part (i) to find the value of \(n\) such that \(\mathrm { P } ( \bar { X } > 5 ) = 0.9377\).
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5 The distance driven in a week by a long-distance lorry driver is a normally distributed random variable with mean 1850 km and standard deviation 117 km .

1. Find the probability that in a random sample of 26 weeks his average distance driven per week is more than 1800 km .
2. New driving regulations are introduced and in a random sample of 26 weeks after their introduction the lorry driver drives a total of 47658 km . Assuming the standard deviation remains unchanged, test at the \(10 \%\) level whether his mean weekly driving distance has changed.

6

1. Explain what is meant by
   1. a Type I error,
   2. a Type II error.
   3. Roger thinks that a box contains 6 screws and 94 nails. Felix thinks that the box contains 30 screws and 70 nails. In order to test these assumptions they decide to take 5 items at random from the box and inspect them, replacing each item after it has been inspected, and accept Roger's hypothesis (the null hypothesis) if all 5 items are nails.
      (a) Calculate the probability of a Type I error.
      (b) If Felix's hypothesis (the alternative hypothesis) is true, calculate the probability of a Type II error.

2 Over a long period of time it is found that the amount of sunshine on any day in a particular town in Spain has mean 6.7 hours and standard deviation 3.1 hours.

1. Find the probability that the mean amount of sunshine over a random sample of 300 days is between 6.5 and 6.8 hours.
2. Give a reason why it is not necessary to assume that the daily amount of sunshine is normally distributed in order to carry out the calculation in part (i).

1 The number of words on a page of a book can be modelled by a normal distribution with mean 403 and standard deviation 26.8. Find the probability that the average number of words per page in a random sample of 6 pages is less than 410.

2 A manufacturer claims that \(20 \%\) of sugar-coated chocolate beans are red. George suspects that this percentage is actually less than \(20 \%\) and so he takes a random sample of 15 chocolate beans and performs a hypothesis test with the null hypothesis \(p = 0.2\) against the alternative hypothesis \(p < 0.2\). He decides to reject the null hypothesis in favour of the alternative hypothesis if there are 0 or 1 red beans in the sample.

1. With reference to this situation, explain what is meant by a Type I error.
2. Find the probability of a Type I error in George's test.

2

1. Write down the mean and variance of the distribution of the means of random samples of size \(n\) taken from a very large population having mean \(\mu\) and variance \(\sigma ^ { 2 }\).
2. What, if anything, can you say about the distribution of sample means
   1. if \(n\) is large,
   2. if \(n\) is small?

5 The length, \(X \mathrm {~cm}\), of a piece of wooden planking is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(b\) is a positive constant.

1. Find the mean and variance of \(X\) in terms of \(b\). The lengths of a random sample of 100 pieces were measured and it was found that \(\Sigma x = 950\).
2. Show that the value of \(b\) estimated from this information is 19 . Using this value of \(b\),
3. find the probability that the length of a randomly chosen piece is greater than 11 cm ,
4. find the probability that the mean length of a random sample of 336 pieces is less than 9 cm .

3 The weights of pebbles on a beach are normally distributed with mean 48.5 grams and standard deviation 12.4 grams.

1. Find the probability that the mean weight of a random sample of 5 pebbles is greater than 51 grams.
2. The probability that the mean weight of a random sample of \(n\) pebbles is less than 51.6 grams is 0.9332 . Find the value of \(n\).

5 The masses of packets of cornflakes are normally distributed with standard deviation 11 g . A random sample of 20 packets was weighed and found to have a mean mass of 746 g .

1. Test at the \(4 \%\) significance level whether there is enough evidence to conclude that the population mean mass is less than 750 g .
2. Given that the population mean mass actually is 750 g , find the smallest possible sample size, \(n\), for which it is at least \(97 \%\) certain that the mean mass of the sample exceeds 745 g .

7

1. Give a reason why sampling would be required in order to reach a conclusion about
   1. the mean height of adult males in England,
   2. the mean weight that can be supported by a single cable of a certain type without the cable breaking.
2. The weights, in kg , of sacks of potatoes are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The weights of a random sample of 500 sacks of potatoes are found and the results are summarised below. $$n = 500 , \quad \Sigma x = 9850 , \quad \Sigma x ^ { 2 } = 194125 .$$
   1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   2. A further random sample of 60 sacks of potatoes is taken. Using your values from part (b) (i), find the probability that the mean weight of this sample exceeds 19.73 kg .
   3. Explain whether it was necessary to use the Central Limit Theorem in your calculation in part (b) (ii).

6 A clinic monitors the amount, \(X\) milligrams per litre, of a certain chemical in the blood stream of patients. For patients who are taking drug \(A\), it has been found that the mean value of \(X\) is 0.336 . A random sample of 100 patients taking a new drug, \(B\), was selected and the values of \(X\) were found. The results are summarised below. $$n = 100 , \quad \Sigma x = 43.5 , \quad \Sigma x ^ { 2 } = 31.56 .$$

1. Test at the \(1 \%\) significance level whether the mean amount of the chemical in the blood stream of patients taking drug \(B\) is different from that of patients taking drug \(A\).
2. For the test to be valid, is it necessary to assume a normal distribution for the amount of chemical in the blood stream of patients taking drug \(B\) ? Justify your answer.

2

1. A random variable \(X\) has mean \(\mu\) and variance \(\sigma ^ { 2 }\). The mean of a random sample of \(n\) values of \(X\) is denoted by \(\bar { X }\). Give expressions for \(\mathrm { E } ( \bar { X } )\) and \(\operatorname { Var } ( \bar { X } )\).
2. The heights, in centimetres, of adult males in Brancot are normally distributed with mean 177.8 and standard deviation 6.1. Find the probability that the mean height of a random sample of 12 adult males from Brancot is less than 176 cm .
3. State, with a reason, whether it was necessary to use the Central Limit Theorem in the calculation in part (ii).

2 A traffic officer notes the speeds of vehicles as they pass a certain point. In the past the mean of these speeds has been \(62.3 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the standard deviation has been \(10.4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). A speed limit is introduced, and following this, the mean of the speeds of 75 randomly chosen vehicles passing the point is found to be \(59.9 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. Making an assumption that should be stated, test at the \(2 \%\) significance level whether the mean speed has decreased since the introduction of the speed limit.
2. Explain whether it was necessary to use the Central Limit theorem in part (i).

5 The number of hours that Mrs Hughes spends on her business in a week is normally distributed with mean \(\mu\) and standard deviation 4.8. In the past the value of \(\mu\) has been 49.5.

1. Assuming that \(\mu\) is still equal to 49.5 , find the probability that in a random sample of 40 weeks the mean time spent on her business in a week is more than 50.3 hours. Following a change in her arrangements, Mrs Hughes wishes to test whether \(\mu\) has decreased. She chooses a random sample of 40 weeks and notes that the total number of hours she spent on her business during these weeks is 1920.
2. (a) Explain why a one-tail test is appropriate.
   (b) Carry out the test at the 6\% significance level.
   (c) Explain whether it was necessary to use the Central Limit theorem in part (ii) (b).

3 The times, in minutes, taken by people to complete a walk are normally distributed with mean \(\mu\). The times, \(t\) minutes, for a random sample of 80 people were summarised as follows. $$\Sigma t = 7220 \quad \Sigma t ^ { 2 } = 656060$$

1. Calculate a \(97 \%\) confidence interval for \(\mu\).
2. Explain whether it was necessary to use the Central Limit theorem in part (i).

2 The mean and standard deviation of the time spent by people in a certain library are 29 minutes and 6 minutes respectively.

1. Find the probability that the mean time spent in the library by a random sample of 120 people is more than 30 minutes.
2. Explain whether it was necessary to assume that the time spent by people in the library is normally distributed in the solution to part (i).