Which one of mean or standard deviation can used to solve the following problem?
A light bulb is considered defective if it lasts less than 400 hours. The following claim is made:
'Brand A light bulbs are more likely to be defective than Brand B light bulbs.'
Is the claim correct?
$$ \begin{array}{c|lc} & \text{} & \text{Mean} & \text{Standard deviation}\\ & Brand A & 450 & 25 \\ & Brand B & 500 & 50 \\ \end{array} $$
My guess is that the claim is incorrect. The reason is that Brand B standard deviation is higher. This shows that although Brand B has a higher mean but the data is more distributed. However I cannot proof my guess. Is there a way that I can find out the number of bulbs that had a higher lifetime of 400, so I can make a comparison?
Assuming that "lamp lasts" distributed normally, we can find, that Z-score for brands A and B are
Z(A) = (450 - 400) / 25 = 2 sigma
Z(B) = (500 - 400) / 50 = 2 sigma
so we can conclude, that probabilities of being defective are the same for both A and B brands; according one tail normal distribution it equals to
P = 0.0228...