The manager of a company that employs 250 travelling sales representatives wishes to carry out a detailed analysis of the expenses claimed by the representatives. He has an alphabetical (by surname) list of the representatives. He chooses a sample of representatives by selecting the 10th, 20th, 30th and so on. Name the type of sampling the manager is attempting to use. Describe a weakness in his method of using it, and explain how he might overcome this weakness.
The representatives each use their own cars to drive to meetings with customers. The total distance, in miles, travelled by a representative in a month is Normally distributed with mean 2018 and standard deviation 96.
Find the probability that, in a randomly chosen month, a randomly chosen representative travels more than 2100 miles.
Find the probability that, in a randomly chosen 3-month period, a randomly chosen representative travels less than 6000 miles. What assumption is needed here? Give a reason why it may not be realistic.
Each month every representative submits a claim for travelling expenses plus commission. Travelling expenses are paid at the rate of 45 pence per mile. The commission is \(10 \%\) of the value of sales in that month. The value, in \(\pounds\), of the monthly sales has the distribution \(\mathrm { N } \left( 21200,1100 ^ { 2 } \right)\). Find the probability that a randomly chosen claim lies between \(\pounds 3000\) and \(\pounds 3300\).
William Sealy, a biochemistry student, is doing work experience at a brewery. One of his tasks is to monitor the specific gravity of the brewing mixture during the brewing process. For one particular recipe, an initial specific gravity of 1.040 is required. A random sample of 9 measurements of the specific gravity at the start of the process gave the following results.
$$\begin{array} { l l l l l l l l l }
1.046 & 1.048 & 1.039 & 1.055 & 1.038 & 1.054 & 1.038 & 1.051 & 1.038
\end{array}$$
William has to test whether the specific gravity of the mixture meets the requirement. Why might a \(t\) test be used for these data and what assumption must be made?
Carry out the test using a significance level of \(10 \%\).
Find a 95\% confidence interval for the true mean specific gravity of the mixture and explain what is meant by a \(95 \%\) confidence interval.