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The concept of the mean, also known as the average, is central to understanding Sally's expenses. It tells us the central tendency or the expected value of a dataset. In this context, it helps us understand Sally's typical daily and weekly expenditures.

To find the mean of her daily expenses, we add the average costs of the individual items she purchases. Since the cost of a coffee averages to $1.40 and a muffin to $2.50, the mean total for each day comes out to $3.90. This tells us that if we look over a period of days, Sally typically spends $3.90 each day on breakfast.

• Mean of Coffee: $1.40
• Mean of Muffin: $2.50
• Mean Total Daily Expense: $1.40 + $2.50 = $3.90

Understanding this mean helps Sally anticipate her expenses over time, ensuring she can budget appropriately for her daily needs.

Standard deviation is a measure of how much variation exists from the average or mean. It's crucial for understanding the consistency of Sally's spending. A low standard deviation means her daily spending is consistently close to the mean, while a high deviation indicates more variability.

For Sally's breakfast, the standard deviation of the coffee's price is \(0.30, and the muffin's is \)0.15. Since these two prices are independent (more on that shortly), we use the following formula to find the total standard deviation for her daily breakfast expenses:\[\text{SD Total Daily Expense} = \sqrt{(\text{SD of Coffee})^2 + (\text{SD of Muffin})^2}\] Plug in the numbers:\[\sqrt{(0.30)^2 + (0.15)^2} \approx 0.335\]Thus, the standard deviation of $0.335 indicates moderate variability in her daily spending.

Independent events are those where the outcome of one does not affect the outcome of another. For Sally, her choice of coffee shop and the prices for coffee and muffins are independent. Each day's prices do not influence the next day's prices.

This independence is significant because it allows us to simply add means and apply the Pythagorean theorem for standard deviations. If these prices were dependent, we would have to consider how the change in one might affect the other.

In practical terms, this means that regardless of what Sally spends today, tomorrow's spending is not affected, so when calculating mean and standard deviation, each day is a new statistical event.

When we refer to Sally’s daily expenses, we're talking about the average amount she spends each day on her breakfast items - coffee and muffin. Understanding daily expenses is important for short-term financial planning.

From our calculations, we know the mean daily expense is $3.90, with a standard deviation of $0.335. This means Sally can expect to spend around $3.90 each day, but should also prepare for some fluctuation. Understanding both mean and standard deviation allows Sally to have a clear picture of her routine spending, ensuring there are no surprises in her daily financial activities.

• Mean Daily Expense: $3.90
• SD Daily Expense: $0.335

Sally's weekly expenses are just her daily expenses multiplied over the seven days of the week. For broader financial planning, Sally benefits from knowing her total expected spending over a longer period.

The mean weekly expense is calculated by multiplying the daily mean of \(3.90 by 7, resulting in \)27.30 for the week. This gives Sally a baseline expectation for her weekly budget. Similarly, we take the daily standard deviation and adjust for a week using the formula \[\text{SD Total Weekly Expense} = \sqrt{7} \times \text{SD Total Daily Expense}\]Plugging in the numbers gives us an approximate weekly standard deviation of $0.886. This standard deviation accounts for the normal fluctuations Sally can expect throughout the week, giving her a full picture of her weekly spending patterns.