Question

in grading of plums whose weights are normally distributed,20% are small, 55% are medium, 15% are large and 10% are very large. if the mean weight of all plums is 5gms with a standard deviation of 1.2 gms what are upper and lower bounds for each category?

Answer 1

Using

$Z=\frac{x_-\mu}{\sigma}\\ x-\mu=\pm z\sigma\\ x=\mu \pm z\sigma\\ For~ small\\ \text{the z value for 0.2 is } \pm 0.842\\ x=5 \pm 0.842(1.2)\\ lower~band=3.9896\\ upper~band=6.0104\\ For~ medium \\ \text{the z value for 0.55 is } \pm 0.126\\ x=5 \pm 0.126(1.2)\\ lower~band=4.8488\\ upper~band=5.1512\\ For~ large\\ \text{the z value for 0.15 is } \pm 1.036\\ x=5 \pm 1.036(1.2)\\ lower~band=3.7568\\ upper~band=6.2432\\ For~very~ large\\ \text{the z value for 0.1 is } \pm 1.282\\ x=5 \pm 1.282(1.2)\\ lower~band=3.4616\\ upper~band=6.5384\\$