A shoe producer makes a variety of​ footwear, including indoor​ slippers, children's​ shoes, and​ flip-flops. To keep up with increasing​ demand, it is considering three expansion​ plans: (1) a small factory with yearly costs of 

​$150 comma 000150,000 

that will increase the production of​ flip-flops to 

400 comma 000400,000​; 

​(2) a​ mid-sized factory with yearly costs of 

​$250 comma 000250,000 

that will increase the production of​ flip-flops to 

600 comma 000600,000​; 

and​ (3) a large factory with yearly costs of 

​$350 comma 000350,000 

that will increase the production of​ flip-flops to 

900 comma 000900,000. 

The profit per​ flip-flop is projected to be 

​$0.750.75. 

The probability distribution of the increased demand for​ flip-flops is provided. Let x represent the amount of profit the producer will make. Complete parts a through c below.

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Part 1

a. Compute the expected profit for each of the expansion plans.

Small Plan 

​E(x)equals=127500127500 

​(Type an integer or a​ decimal.)

Part 2

​Mid-Sized Plan 

​E(x)equals=147500147500 

​(Type an integer or a​ decimal.)

Part 3

Large Plan 

​E(x)equals=152500152500 

​(Type an integer or a​ decimal.)

Part 4

b. Calculate the standard deviation for each of the expansion plans.

Small Plan 

sigmaσ Subscript xequals=


Let's break down the steps to calculate the standard deviation for each of the expansion plans, assuming we have the probability distribution for the increased demand of flip-flops.

### Step 1: Review Expected Value Formula
The expected value (mean) $ E(x) $ is given by:

\[
E(x) = \sum (x_i \cdot p_i)
\]

Where:
- $ x_i $ is the profit for a particular demand level.
- $ p_i $ is the probability of that particular demand.

### Step 2: Standard Deviation Formula
The formula for standard deviation $ \sigma_x $ is:

\[
\sigma_x = \sqrt{\sum \left( (x_i - E(x))^2 \cdot p_i \right)}
\]

Where:
- $ E(x) $ is the expected value (profit), which has already been computed for each plan.
- $ x_i $ represents the profit values for different demand levels.
- $ p_i $ is the corresponding probability of each demand level.

To calculate the standard deviation for each expansion plan, we need:
1. The demand levels and their probabilities.
2. The formula to calculate the variance by plugging in the expected value $ E(x) $, the profit values $ x_i $, and the probabilities $ p_i $.

Please share the probability distribution of the increased demand, and I will help you compute the standard deviation.

Would you like a detailed step-by-step explanation on how to compute the standard deviation for each plan?

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### Five Related Questions:
1. How is the expected profit for each plan affected by changes in demand probability?
2. How would changing the cost structure (e.g., increasing the yearly costs) impact the expected profit and risk?
3. What happens to the standard deviation if the probabilities of lower demand increase?
4. How can we interpret the standard deviation in terms of business risk?
5. What other factors (like market conditions) might affect the choice of expansion plan?

### Tip:
Always consider both expected profit and standard deviation when making business decisions to balance potential profit with associated risk.

A shoe producer makes a variety of footwear, including indoor slippers, children's shoes, and flip-flops. To keep up with increasing demand, it is considering three expansion plans: (1) a small factory with yearly costs of $150,000 that will increase the production of flip-flops to 400,000; (2) a mid-sized factory with yearly costs of $250,000 that will increase the production of flip-flops to 600,000; and (3) a large factory with yearly costs of $350,000 that will increase the production of flip-flops to 900,000. The profit per flip-flop is projected to be $0.75. The probability distribution of the increased demand for flip-flops is provided. Let x represent the amount of profit the producer will make. Compute the expected profit and the standard deviation for each of the expansion plans.

Calculate the expected profit and standard deviation for a shoe producer considering expansion plans. The problem involves three different factory options, expected profits, and a probability distribution of demand. Suitable for college-level business or finance students.

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