The distribution of egg weights (g) of a certain type is normal with
- mean value 53
- standard deviation .3
- (consistent with data in the article “Evaluation of Egg Quality Traits of Chickens Reared under Backyard System in Western Uttar Pradesh” (Indian J. of Poultry Sci., 2009: 261-262)). Let denote the weights of a dozen randomly selected eggs;
- these ’s constitute a random sample of size 12 from the specified normal distribution.
The total weight of the 12 eggs is ; it is normally distributed with
- mean value
- variance 1.08.
The probability that the total weight is between 635 and 640 is now obtained by standardizing and referring to Appendix Table A.3:
If cartons containing a dozen eggs are repeatedly selected, in the long run slightly more than of the eggs in a carton will weigh in total between and . Notice that is equivalent to (divide each term in the original system of inequalities by 12). Thus .8315. This latter probability can also be obtained by standardizing directly.
Now consider randomly selecting just four of these eggs. The sample mean weight is then normally distributed with mean value and standard deviation . The probability that the sample mean weight exceeds is then
Because 53.5 is 3.33 standard deviations (of ) larger than the mean value 53, it is exceedingly unlikely that the sample mean will exceed 53.5.