Single normal population sample mean

2 The time, in minutes, taken by students to complete a test has the distribution \(\mathrm { N } ( 125,36 )\).

1. Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123 minutes.
2. Explain whether it was necessary to use the Central Limit theorem in the solution to part (a).

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 .

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 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 \(\mathrm { P } ( \bar { X } > 64 )\).

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.

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.

3 A men's triathlon consists of three parts: swimming, cycling and running. Competitors' times, in minutes, for the three parts can be modelled by three independent normal variables with means 34.0, 87.1 and 56.9, and standard deviations 3.2, 4.1 and 3.8, respectively. For each competitor, the total of his three times is called the race time. Find the probability that the mean race time of a random sample of 15 competitors is less than 175 minutes.

5 A food company makes mini apple pies. The weight of pastry in a pie is Normally distributed with mean 75 g and standard deviation 4 g . The weight of filling in a pie is Normally distributed with mean 130 g and standard deviation 8 g . You should assume that the weights of pastry and filling in a pie are independent.

1. Find the probability that the weight of pastry in a randomly chosen pie is between 70 g and 80 g .
2. Find the probability that the mean weight of filling in 10 randomly chosen pies is at least 125 g. The pies are sold in packs of 4 . The weight of the packaging is Normally distributed with mean 165 g and standard deviation 6 g .
3. In order to find the probability that the total weight of a pack of 4 pies is less than 1 kg , you must assume that the weight of the packaging is independent of the weight of the pies.
   1. State another necessary assumption.
   2. Given that the assumptions are valid, calculate this probability.