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An order for bottles of vitamins from a certain mail order company costs $\$ 12.04$ per bottle plus a shipping cost of $\$ 4.80$ regardless of the number of bottles ordered. Over the past year, the company has received 100 orders for bottles of vitamins, The standard deviation of the numbers of bottles per order for the 100 orders is 1.5 bottles. What is the standard deviation of the 100 costs for the orders?

The mean number of bottles per order is unknown.

The total number of bottles sold is unknown.

The total number of bottles equals $100 \ \cdot$ mean number of bottles per order.

$$\sqrt{ \dfrac{\sum_{i=1}^{100} (bottles_{ith \ order}-\dfrac{\sum bottles}{orders})^2}{100}}= standard \ deviation =\sqrt{\dfrac{ \sum_{i=1}^{100}(bottles_{ith \ order}-\dfrac{\sum bottles}{100})^2}{100}}=1.5$$.

$12.04 \sum bottles+4.8 \cdot orders=12.04 \sum bottles+4.8 \cdot 100 = total \ cost$

$\sum bottles = \dfrac{total \ cost - 4.8 \cdot 100}{12.04}$

How to proceed?

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1 Answer 1

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In general, multiplying every observation by a positive constant multiplies the standard deviation by that constant, and adding a constant to every observation doesn't change the standard deviation. So here the standard deviation is $12.04 \cdot 1.5 = \$18.06$.

To be a bit more formal about it: Let $B$ be the number of bottles in an order. We know $\text{Var}(B)=1.5^2$ and we want to know:

$$\text{Var}(12.04B + 4.80) = \text{Var}(12.04B)=12.04^2\text{Var}(B)=12.04^2\cdot1.5^2$$

Then the standard deviation is $\sqrt{\text{Var}(B)}=\sqrt{12.04^2\cdot1.5^2} = 12.04 \cdot 1.5 = \boxed{18.06}$.

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