91Ó°ÊÓ

"Macaroni and Cheese, please!!" by Nedda Misherghi and Rachelle Hall As a poor starving student I don't have much money to spend for even the bare necessities. So my favorite and main staple food is macaroni and cheese. It's high in taste and low in cost and nutritional value. One day, as I sat down to determine the meaning of life, I got a serious craving for this, oh, so important, food of my life. So I went down the street to Great way to get a box of macaroni and cheese, but it was SO expensive! 2.02 dollar !!! Can you believe it? It made me stop and think. The world is changing fast. I had thought that the mean cost of a box (the normal size, not some super-gigantic- family-value-pack) was at most 1 dollar, but now I wasn't so sure. However, I was determined to find out. I went to 53 of the closest grocery stores and surveyed the prices of macaroni and cheese. Here are the data I wrote in my notebook: Price per box of Mac and Cheese: \- 5 stores @ 2.02 dollar \- 15 stores @ 0.25 dollar \- 3 stores @ 1.29 dollar \- 6 stores @ 0.35 dollar \- 4 stores @ 2.27 dollar \- 7 stores @ 1.50 \- 5 stores @ 1.89 dollar \- 8 stores @ 0.75 . I could see that the cost varied but I had to sit down to figure out whether or not I was right. If it does turn out that this mouth-watering dish is at most 1 dollar, then I'll throw a big cheesy party in our next statistics lab, with enough macaroni and cheese for just me. (After all, as a poor starving student I can't be expected to feed our class of animals!)

Step by step solution

01

Consolidate the Data

We start by writing down the number of stores and their corresponding prices: - 5 stores at $2.02 - 15 stores at $0.25 - 3 stores at $1.29 - 6 stores at $0.35 - 4 stores at $2.27 - 7 stores at $1.50 - 5 stores at $1.89 - 8 stores at $0.75.

02

Calculate the Total Cost

Calculate the total cost of macaroni and cheese by multiplying the number of stores by the price at which they sell:\( 5 \times 2.02 + 15 \times 0.25 + 3 \times 1.29 + 6 \times 0.35 + 4 \times 2.27 + 7 \times 1.50 + 5 \times 1.89 + 8 \times 0.75 \).

03

Perform the Multiplication

Calculate each product:\(5 \times 2.02 = 10.10,15 \times 0.25 = 3.75,3 \times 1.29 = 3.87,6 \times 0.35 = 2.10,4 \times 2.27 = 9.08,7 \times 1.50 = 10.50,5 \times 1.89 = 9.45,8 \times 0.75 = 6.00\).

04

Sum the Total Costs

Now, add the products from Step 3: \(10.10 + 3.75 + 3.87 + 2.10 + 9.08 + 10.50 + 9.45 + 6.00\). This gives us a total cost of \(54.85\) dollars.

05

Calculate the Mean Cost

To find the mean cost, divide the total cost by the total number of observations (53 stores): \(\frac{54.85}{53} = 1.035\).

06

Interpret the Mean Cost

The mean cost of a box of macaroni and cheese is \(1.035\) dollars, which is greater than 1 dollar.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistics Problem Solving

When solving a statistics problem involving mean cost calculation, it's important to follow a structured approach. The situation above revolves around figuring out if the mean cost of a staple food, macaroni and cheese, exceeds a specific threshold – in this case, 1 dollar. This scenario is a typical example of a statistics application where a numerical analysis is needed.

To start, you must comprehend the presented data. Notice how each store offers macaroni and cheese at different prices. The target is to compile this data logically and calculate an average, or mean. Mean is a summary statistic, an estimation of the central tendency of prices, which indicates the possible average price you might expect to pay across different grocery stores. By breaking down this problem into manageable steps, we can clearly elucidate the steps to find this mean effectively.

Using statistics for such everyday decisions exemplifies how math can help us make informed economic choices in life. Calculating mean costs can guide budget decisions, especially if you're trying to determine whether pricing trends align with your financial expectations or past experiences.

Data Consolidation

Data consolidation is a fundamental process needed before performing any statistical analysis. It's the act of grouping and summarizing data to make it manageable. In this exercise, before calculating the mean cost of macaroni and cheese, it's crucial to record the prices accurately.

You'll want to list the number of stores and their respective costs of selling macaroni and cheese in a structured way. For instance:

• 5 stores selling at $2.02
• 15 stores selling at $0.25
• ... and so forth.

Each price point is tied directly to a corresponding number of stores, consolidating this data clarifies the scope of our analysis.

The next step is performing operations that include multiplying to find total costs per price point, as shown in the problem's solution method. Consolidating data in this manner simplifies complex datasets into comprehensive, actionable information. It prepares you for the analytical operations that follow, ensuring accuracy in calculating output parameters like total cost and mean price.

Average Price Analysis

The concept of average price analysis involves finding the mean price from a set of data. To do this efficiently, begin by calculating the total cost spent on macaroni and cheese across all stores surveyed. This total cost represents the price paid if one had bought the product at each listed price proportionally.

In our example, every store's price is considered by multiplying the price by the number of stores offering that price, and then all these results are summed up to get a grand total cost.

• Calculate each contribution: For example, 5 stores at $2.02 translates to a total of $10.10 for those particular 5 stores.
• Add these values together to determine the total outlay: Here, sums like $10.10 + $3.75 + ... accumulate to $54.85.

Once this is done, you can then calculate the mean by dividing this total by the total number of observations, i.e., stores (53 in our example).

After computation, if the result exceeds a dollar, then you'd know the average cost does not align with your initial expectation of $1. By employing average price analysis, we gain valuable insights into regional cost tendencies of products like macaroni and cheese, which can confirm or challenge previous assumptions about pricing.