The length of one-metre galvanised-steel straps used in house building may be modelled by a normal distribution with a mean of 1005 mm and a standard deviation of 15 mm .
The straps are supplied to house builders in packs of 12, and the straps in a pack may be assumed to be a random sample.
Determine the probability that the mean length of straps in a pack is less than one metre.
Tania, a purchasing officer for a nationwide house builder, measures the thickness, \(x\) millimetres, of each of a random sample of 24 galvanised-steel straps supplied by a manufacturer. She then calculates correctly that the value of \(\bar { x }\) is 4.65 mm .
Assuming that the thickness, \(X \mathrm {~mm}\), of such a strap may be modelled by the distribution \(\mathrm { N } \left( \mu , 0.15 ^ { 2 } \right)\), construct a \(99 \%\) confidence interval for \(\mu\).
Hence comment on the manufacturer's specification that the mean thickness of such straps is greater than 4.5 mm .