3 A builders' merchant's depot has two machines, X and Y , each of which can be used for filling bags with sand or gravel.
The weight, in kilograms, delivered by machine X may be modelled by a normal distribution with mean \(\mu _ { \mathrm { X } }\) and standard deviation 25 .
The weight, in kilograms, delivered by machine Y may be modelled by a normal distribution with mean \(\mu _ { \mathrm { Y } }\) and standard deviation 30 .
Fred, the depot's yardman, records the weights, in kilograms, of a random sample of 10 bags of sand delivered by machine X as
\(\begin{array} { l l l l l l l l l l } 1055 & 1045 & 1000 & 985 & 1040 & 1025 & 1005 & 1030 & 1015 & 1060 \end{array}\)
He also records the weights, in kilograms, of a random sample of 8 bags of gravel delivered by machine Y as
$$\begin{array} { l l l l l l l l }
1085 & 1055 & 1055 & 1000 & 1035 & 1050 & 1005 & 1075
\end{array}$$
- Construct a \(95 \%\) confidence interval for \(\mu _ { \mathrm { Y } } - \mu _ { \mathrm { X } }\), giving the limits to the nearest 5 kg .
- Dot, the depot's manager, commented that Fred's data collection may have been biased.
Justify her comment and explain how the possible bias could have been eliminated.
(2 marks)