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Understanding mean calculation is crucial for analyzing daily expenses. In this exercise, we found Sally's mean daily cost for breakfast by adding the average price of her coffee and muffin.

• The average price of a cup of coffee is \(1.40.
• The average price of a muffin is \)2.50.

To calculate the mean cost, we simply add these two amounts: \[\text{Mean Daily Cost} = 1.40 + 2.50 = 3.90\] Therefore, Sally spends on average $3.90 each day for her breakfast. It's important to note that the mean gives us a central value around which other values are distributed, making it a helpful summary statistic.

Standard deviation tells us how much variation or spread there is in a set of values. A lower standard deviation implies that the values are close to the mean, while a higher standard deviation indicates more variation.
When calculating the standard deviation for the total cost of independent expenses (like Sally's coffee and muffin), we use the formula:\[\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}\]Here, \(\sigma_1 = 0.30\) for the coffee and \(\sigma_2 = 0.15\) for the muffin. Calculating these gives us:\[\sqrt{0.30^2 + 0.15^2} = \sqrt{0.1125} \approx 0.335\]This standard deviation of $0.335 tells us the amount of variation in her daily expense.

Independent random variables are those whose outcomes do not affect each other. In Sally's case, the prices of coffee and muffins are independent.
This means the variation in coffee prices does not influence muffin prices and vice versa. When calculating the standard deviation for the total expense, we leveraged this independence.

• This property allows us to sum the variances (square of standard deviations) when finding the overall standard deviation.
• If they were not independent, the calculation would require additional correlation information.

Understanding the concept of independence helps in accurate statistical analysis, especially when combining different random variables.

Cost analysis provides insights into spending habits over a defined period. By analyzing both daily and weekly costs, you can identify patterns and budget effectively.
In Sally's case, we found her weekly expected costs by multiplying her daily average by 7 (the number of days in a week):\[\text{Mean Weekly Cost} = 7 \times 3.90 = 27.30\]Similarly, to find her weekly standard deviation, we multiply the daily standard deviation by the square root of 7:\[\text{Weekly Standard Deviation} = \sqrt{7} \times 0.335 \approx 0.887\]This analysis reveals that over a week, Sally's expenses are not just a linear expansion from daily expenses as the variability also increases, reflecting more possible fluctuation in total spending.