5.05a Sample mean distribution: central limit theorem

1 A random variable \(X\) has the distribution \(\mathrm { N } ( 410,400 )\).
Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405 .

2 The length, in minutes, of mathematics lectures at a certain college has mean \(\mu\) and standard deviation 8.3.

1. The total length of a random sample of 85 lectures was 4590 minutes. Calculate a 95\% confidence interval for \(\mu\).
   The length, in minutes, of history lectures at the college has mean \(m\) and standard deviation \(s\).
2. Using a random sample of 100 history lectures, a 95\% confidence interval for \(m\) was found to have width 2.8 minutes. Find the value of \(s\).

4 The height \(H\), in metres, of mature trees of a certain variety is normally distributed with standard deviation 0.67. In order to test whether the population mean of \(H\) is greater than 4.23, the heights of a random sample of 200 trees are measured.

1. Write down suitable null and alternative hypotheses for the test.
   The sample mean height, \(\bar { h }\) metres, of the 200 trees is found and the test is carried out. The result of the test is to reject the null hypothesis at the 5\% significance level.
2. Find the set of possible values of \(\bar { h }\).
3. Ajit said, 'In (b) we had to assume that \(\bar { H }\) is normally distributed, so it was necessary to use the Central Limit Theorem.' Explain whether you agree with Ajit.

3 The times, \(T\) minutes, taken by a random sample of 75 students to complete a test were noted. The results were summarised by \(\Sigma t = 230\) and \(\Sigma t ^ { 2 } = 930\).

1. Calculate unbiased estimates of the population mean and variance of \(T\).
   You should now assume that your estimates from part (a) are the true values of the population mean and variance of \(T\).
2. The times taken by another random sample of 75 students were noted, and the sample mean, \(\bar { T }\), was found. Find the value of \(a\) such that \(P ( \bar { T } > a ) = 0.234\).

4 A population is normally distributed with mean 35 and standard deviation 8.1 . A random sample of size 140 is chosen from this population and the sample mean is denoted by \(\bar { X }\).

1. Find \(\mathrm { P } ( \bar { X } > 36 )\).
2. It is given that \(\mathrm { P } ( \bar { X } < a ) = 0.986\). Find the value of \(a\).

3 Random samples of size 120 are taken from the distribution \(\mathrm { B } ( 15,0.4 )\).

1. Describe fully the distribution of the sample mean.
2. Find the probability that the mean of a random sample of size 120 is greater than 6.1.

1 The random variable \(X\) has the distribution \(\mathrm { B } ( 10,0.15 )\). Find the probability that the mean of a random sample of 50 observations of \(X\) is greater than 1.4.

2 The lengths of time people take to complete a certain type of puzzle are normally distributed with mean 48.8 minutes and standard deviation 15.6 minutes. The random variable \(X\) represents the time taken in minutes by a randomly chosen person to solve this type of puzzle. The times taken by random samples of 5 people are noted. The mean time \(\bar { X }\) is calculated for each sample.

1. State the distribution of \(\bar { X }\), giving the values of any parameters.
2. Find \(\mathrm { P } ( \bar { X } < 50 )\).

3 Metal bolts are produced in large numbers and have lengths which are normally distributed with mean 2.62 cm and standard deviation 0.30 cm .

1. Find the probability that a random sample of 45 bolts will have a mean length of more than 2.55 cm .
2. The machine making these bolts is given an annual service. This may change the mean length of bolts produced but does not change the standard deviation. To test whether the mean has changed, a random sample of 30 bolts is taken and their lengths noted. The sample mean length is \(m \mathrm {~cm}\). Find the set of values of \(m\) which result in rejection at the \(10 \%\) significance level of the hypothesis that no change in the mean length has occurred.

3 The weight, in grams, of a certain type of apple is modelled by the random variable \(X\) with mean 62 and standard deviation 8.2. A random sample of 50 apples is selected, and the mean weight in grams, \(\bar { X }\), is found.

1. Describe fully the distribution of \(\bar { X }\).
2. Find \(\mathrm { P } ( \bar { X } > 64 )\).

7 Previous records have shown that the number of cars entering Bampor on any day has mean 352 and variance 121.

1. Find the probability that the mean number of cars entering Bampor during a random sample of 200 days is more than 354 .
2. State, with a reason, whether it was necessary to assume that the number of cars entering Bampor on any day has a normal distribution in order to find the probability in part (i).
3. It is thought that the population mean may recently have changed. The number of cars entering Bampor during the day was recorded for each of a random sample of 50 days and the sample mean was found to be 356 . Assuming that the variance is unchanged, test at the \(5 \%\) significance level whether the population mean is still 352 .

2 A population has mean 7 and standard deviation 3. A random sample of size \(n\) is chosen from this population.

1. Write down the mean and standard deviation of the distribution of the sample mean.
2. Under what circumstances does the sample mean have
   1. a normal distribution,
   2. an approximately normal distribution?

3 The lengths, \(x \mathrm {~mm}\), of a random sample of 150 insects of a certain kind were found. The results are summarised by \(\Sigma x = 7520\) and \(\Sigma x ^ { 2 } = 413540\).

1. Calculate unbiased estimates of the population mean and variance of the lengths of insects of this kind.
2. Using the values found in part (i), calculate an estimate of the probability that the mean length of a further random sample of 80 insects of this kind is greater than 53 mm .

4 The lengths, \(x \mathrm {~m}\), of a random sample of 200 balls of string are found and the results are summarised by \(\Sigma x = 2005\) and \(\Sigma x ^ { 2 } = 20175\).

1. Calculate unbiased estimates of the population mean and variance of the lengths.
2. Use the values from part (i) to estimate the probability that the mean length of a random sample of 50 balls of string is less than 10 m .
3. Explain whether or not it was necessary to use the Central Limit theorem in your calculation in part (ii).

7 In the past the weekly profit at a store had mean \(\\) 34600\( and standard deviation \)\\( 4500\). Following a change of ownership, the mean weekly profit for 90 randomly chosen weeks was \(\\) 35400$.

1. Stating a necessary assumption, test at the \(5 \%\) significance level whether the mean weekly profit has increased.
2. State, with a reason, whether it was necessary to use the Central Limit theorem in part (i). The mean weekly profit for another random sample of 90 weeks is found and the same test is carried out at the 5\% significance level.
3. State the probability of a Type I error.
4. Given that the population mean weekly profit is now \(\\) 36500$, calculate the probability of a Type II error.

5 The score on one throw of a 4 -sided die is denoted by the random variable \(X\) with probability distribution as shown in the table.

1. Show that \(\operatorname { Var } ( X ) = 1.25\). The die is thrown 300 times. The score on each throw is noted and the mean, \(\bar { X }\), of the 300 scores is found.
2. Use a normal distribution to find \(\mathrm { P } ( \bar { X } < 1.4 )\).
3. Justify the use of the normal distribution in part (ii).

3 The daily times, in minutes, that Yu Ming takes showering, getting dressed and having breakfast are independent and have the distributions \(\mathrm { N } \left( 9,2.2 ^ { 2 } \right) , \mathrm { N } \left( 8,1.3 ^ { 2 } \right)\) and \(\mathrm { N } \left( 17,2.6 ^ { 2 } \right)\) respectively. The total daily time that Yu Ming takes for all three activities is denoted by \(T\) minutes.

1. Find the mean and variance of \(T\).
2. Yu Ming notes the value of \(T\) on each day in a random sample of 70 days and calculates the sample mean. Find the probability that the sample mean is between 33 and 35 .

5 The mass, in kilograms, of rocks in a certain area has mean 14.2 and standard deviation 3.1.

1. Find the probability that the mean mass of a random sample of 50 of these rocks is less than 14.0 kg .
2. Explain whether it was necessary to assume that the population of the masses of these rocks is normally distributed.
3. A geologist suspects that rocks in another area have a mean mass which is less than 14.2 kg . A random sample of 100 rocks in this area has sample mean 13.5 kg . Assuming that the standard deviation for rocks in this area is also 3.1 kg , test at the \(2 \%\) significance level whether the geologist is correct.

3 The management of a factory wished to find a range within which the time taken to complete a particular task generally lies. It is given that the times, in minutes, have a normal distribution with mean \(\mu\) and standard deviation 6.5. A random sample of 15 employees was chosen and the mean time taken by these employees was found to be 52 minutes.

1. Calculate a \(95 \%\) confidence interval for \(\mu\).
   Later another \(95 \%\) confidence interval for \(\mu\) was found, based on a random sample of 30 employees.
2. State, with a reason, whether the width of this confidence interval was less than, equal to or greater than the width of the previous interval.

4 The mean mass of packets of sugar is supposed to be 505 g . A random sample of 10 packets filled by a certain machine was taken and the masses, in grams, were found to be as follows. $$\begin{array} { l l l l l l l l l l } 500 & 499 & 496 & 495 & 498 & 490 & 492 & 501 & 494 & 494 \end{array}$$

1. Find unbiased estimates of the population mean and variance.
   The mean mass of packets produced by this machine was found to be less than 505 g , so the machine was adjusted. Following the adjustment, the masses of a random sample of 150 packets from the machine were measured and the total mass was found to be 75660 g .
2. Given that the population standard deviation is 3.6 g , test at the \(2 \%\) significance level whether the machine is still producing packets with mean mass less than 505 g .
3. Explain why the use of the normal distribution is justified in carrying out the test in part (ii). [1]

2 The time, in minutes, that John takes to travel to work has a normal distribution. Last year the mean and standard deviation were 26.5 and 4.8 respectively. This year John uses a different route and he finds that the mean time for his first 150 journeys is 27.5 minutes.

1. Stating a necessary assumption, test at the \(1 \%\) significance level whether the mean time for his journey to work has increased.
2. State, with a reason, whether it was necessary to use the Central Limit theorem in your answer to part (i).

5

1. The random variable \(X\) has the distribution \(\operatorname { Po } ( 2.3 )\).
   1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
   2. Find the probability that the sum of two independent values of \(X\) is greater than 2 .
   3. The random variable \(S\) is the sum of 50 independent values of \(X\). Use a suitable approximating distribution to find \(\mathrm { P } ( S \leqslant 110 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\). Given that \(\mathrm { P } ( Y = 3 ) = \mathrm { P } ( Y = 5 )\), find \(\lambda\).

2 The random variable \(X\) has mean 372 and standard deviation 54 .

1. Describe fully the distribution of the mean of a random sample of 36 values of \(X\).
2. The distribution in part (i) might be either exact or approximate. State a condition under which the distribution is exact.

3 In the past, Arvinder has found that the mean time for his journey to work is 35.2 minutes. He tries a different route to work, hoping that this will reduce his journey time. Arvinder decides to take a random sample of 25 journeys using the new route. If the sample mean is less than 34.7 minutes he will conclude that the new route is quicker. Assume that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.

1. Find the probability that a Type I error occurs.
2. Arvinder finds that the sample mean is 34.5 minutes. Explain briefly why it is impossible for him to make a Type II error.

3 The length, in centimetres, of a certain type of snake is modelled by the random variable \(X\) with mean 52 and standard deviation 6.1. A random sample of 75 snakes is selected, and the sample mean, \(\bar { X }\), is found.

1. Find \(\mathrm { P } ( 51 < \bar { X } < 53 )\).
2. Explain why it was necessary to use the Central Limit theorem in the solution to part (i).