Chapter 18, Problem 1E

Salmon A specialty food company sells whole King Salmon to various customers. The mean weight of these salmon is 35 pounds with a standard deviation of 2 pounds. The company ships them to restaurants in boxes of 4 salmon, to grocery stores in cartons of 16 salmon, and to discount outlet stores in pallets of 100 salmon. To forecast costs, the shipping department needs to estimate the standard deviation of the mean weight of the salmon in each type of shipment.

1. a) Find the standard deviations of the mean weight of the salmon in each type of shipment.
2. b) The distribution of the salmon weights turns out to be skewed to the high end. Would the distribution of shipping weights be better characterized by a Normal model for the boxes or pallets? Explain.

To determine

Find the standard deviations of the mean weight of the salmon in each type of shipment.

Answer to Problem 1E

The standard deviations are 1 pound, 0.5 pound and 0.2 pounds.

Explanation of Solution

Given info:

A food company sells King Salmon to customers. The mean weight of these salmon is 35 pounds and standard deviation is 2 pounds. They delivered three different restaurants in boxes with 4 salmon, in cartons 16 salmon and in pallets 100 salmon respectively.

Calculation:

Standard deviation:

For shipment of 4 salmon the standard deviation is,

$SD=\frac{\sigma }{\sqrt{n}}=\frac{2}{\sqrt{4}}=1$

For shipment of 16 salmon the standard deviation is,

$SD=\frac{\sigma }{\sqrt{n}}=\frac{2}{\sqrt{16}}=0.5$

For shipment of 100 salmon the standard deviation is,

$SD=\frac{\sigma }{\sqrt{n}}=\frac{2}{\sqrt{100}}=0.2$

Thus, the standard deviations are 1 pound, 0.5 pound and 0.2 pounds for 4 salmons, 16 salmons and 100 salmons respectively.

To determine

Explain whether the distribution of shipping weights be better characterized by a normal model for the boxes or pallets or not.

Explanation of Solution

Given info:

The distribution of the salmon weights turns out to be skewed to the high end.

Justification:

By the central limit theorem, it can be said that for large sample size the distribution of the population mean follows normal distribution.

As pallets contains large number of salmon than the boxes. Hence, the distribution of shipping weights is better characterized by a Normal model for the pallets.