Question

A farmer has 100 lb of apples and 50 lb of potatoes for sale. The market price for apples (per pound) each day is a random variable with a mean of 0.5 dollars and a standard deviation of 0.2 dollars. Similarly, for a pound of potatoes, the mean price is 0.3 dollars and the standard deviation is 0.1 dollars. It also costs him 2 dollars to bring all the apples and potatoes to the market. The market is busy with eager shoppers, so we can assume that he'll be able to sell all of each type of produce at that day's price. Do you need to make any assumptions in calculating the mean? How about the standard deviation?

Answer 1

The average earnings from selling apples and potatoes, after deducting transportation costs, is $63. To find the standard deviation of earnings, we must consider the given standard deviations of apple and potato prices and calculate using the formula for independent variables.

Step-by-step explanation:

The student's question is about calculating the expected mean earnings and the assessment of variance (standard deviation) from selling apples and potatoes at a farmers market given the mean and standard deviation of the market prices. To calculate the mean earnings, we make the assumption that prices for apples and potatoes are independent random variables. The mean earnings for apples would be 100 lb × $0.50 = $50.00, and for potatoes, it would be 50 lb × $0.30 = $15.00. After subtracting the cost of bringing produce to the market, the total mean earnings would be $50 + $15 - $2 = $63.00.

When considering standard deviation, since the prices of apples and potatoes are independent, we use the formula for the standard deviation of independent variables (which involves squaring the standard deviations, multiplying by the square of the quantity of each produce, summing them, and then taking the square root). We would not add the standard deviations directly but rather use this formula to assess the overall risk or variability in the farmer's potential earnings.

Answer 2

To calculate the mean and standard deviation for the market prices of apples and potatoes, we need to assume that the prices are normally distributed and that they are independent of each other.

Step-by-step explanation:

When calculating the mean and standard deviation for the market price of apples and potatoes, we have to make certain assumptions. In this case, we need to assume that the market prices for apples and potatoes are normally distributed. This means that the prices are spread out symmetrically around the mean, and most prices fall close to the mean. Additionally, we assume that the prices for apples and potatoes are independent of each other, meaning that the price of one does not affect the price of the other.

Regarding the mean, we can calculate it by taking the average of all the possible prices for apples and potatoes. The standard deviation can be calculated by determining the spread or variability of prices around the mean.