Sample size determination

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 \(n\).

2 The lengths of sewing needles in travel sewing kits are distributed normally with mean \(\mu \mathrm { mm }\) and standard deviation 1.5 mm . A random sample of \(n\) needles is taken. Find the smallest value of \(n\) such that the width of a \(95 \%\) confidence interval for the population mean is at most 1 mm .

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 \(n\).

Aram is a packer at a supermarket checkout and the time he takes to pack a randomly chosen item has mean 1.5 s and standard deviation 0.4 s . Justifying any approximation that you make, find the probability that Aram will pack 50 randomly chosen items in less than 70 s . Find the greatest number of items that Aram could pack within 70 s with probability at least \(90 \%\). Huldu is also a packer at the supermarket. The time that she takes to pack a randomly chosen item has mean 1.3 s and standard deviation 0.5 s . Aram and Huldu each have 50 items to pack. Find the probability that Huldu takes a shorter time than Aram.

6 Population \(A\) has a normal distribution with unknown mean \(\mu _ { A }\) and a variance of 18.8.
Population \(B\) has a normal distribution with unknown mean \(\mu _ { B }\) but with the same variance as Population \(A\). The random variables \(\bar { X } _ { A }\) and \(\bar { X } _ { B }\) denote the means of independent samples, each of size \(n\), from populations \(A\) and \(B\) respectively.

1. Find an expression, in terms of \(n\), for \(\operatorname { Var } \left( \bar { X } _ { A } - \bar { X } _ { B } \right)\).
2. Given that the width of a \(99 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\) is to be at most 5 , calculate the minimum value for \(n\).
   [0pt] [5 marks]

5. The masses, \(X\), in kg, of men who work for a large company are normally distributed with mean 75 and standard deviation 10.

1. Find the probability that the mean mass of a random sample of 5 men is less than 70 kg .
2. The mean mass, in kg , of a random sample of \(n\) men drawn from this distribution is \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } > 80 )\) is approximately \(0 \cdot 007\), find \(n\). The masses, in kg, of women who work for the company are normally distributed with mean 68 and standard deviation 6 . A lift in the company building will not move if the total mass in the lift is more than 500 kg .
3. A random sample of 3 men and 4 women get in the lift. Find the probability that the lift will not move.
4. State a modelling assumption you have made in calculating your answer for part (c).