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Imagine you have a very large coin jar full of nickels, dimes, and quarters. You would like to know how much money you have in the jar, but you don't want to go through the trouble of counting all the coins. You decide to estimate how many nickels, dimes, and quarters are in the jar using the capture- recapture method. After shaking the jar well, you draw a first sample of 150 coins and get 36 quarters, 45 nickels, and 69 dimes. Using a permanent ink marker you tag each of the 150 coins with a black dot and put the coins back in the jar, shake the jar really well to let the tagged coins mix well with the rest, and draw a second sample of 100 coins. The second sample has 28 quarters, 29 nickels, and 43 dimes. Of these, 4 quarters, 5 nickels, and 8 dimes have black dots. Estimate how much money is in the jar. (Hint: You will need a separate calculation for estimating the quarters, nickels, and dimes in the jar.)

Step by step solution

01

Estimate the total number of each type of coin

Based on the capture-recapture method, the total number of each type of coin can be estimated using the formula: \((\text{Total number of first sample} \times \text{Total number of second sample}) / \text{Number of recaptured coins}\). For example, the estimated total number of quarters is \((150 \times 28) / 4\). So the same should be done for nickels and dimes.

02

Calculate monetary value of estimated coins

Once the estimated total number of each type of coin is calculated, convert these to their monetary values. For quarters, the value is \(\text{Estimated Quarters} \times 0.25\). For dimes, the value is \(\text{Estimated Dimes} \times 0.10\). And for nickels, the value is \(\text{Estimated Nickels} \times 0.05\).

03

Sum the estimates

Finally, sum up the calculated monetary values of quarters, dimes, and nickels to estimate the total amount of money in the jar. This can be calculated as \(\text{Monetary Value of Quarters} + \text{Monetary Value of Dimes} + \text{Monetary Value of Nickels}\).

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Key Concepts

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

Coin Estimation

When faced with a situation where counting each individual coin is impractical, estimation becomes a valuable tool. In this exercise, we aim to estimate the number of each coin type—nickels, dimes, and quarters—in a large jar using the capture-recapture method. This technique originates from ecology where it helps estimate animal populations. Here's a brief run-through:

• Firstly, we take an initial sample. In this case, it consisted of 150 coins, recording how many of each type were drawn.
• Next, we mark these sampled coins distinctly and return them to the jar, ensuring they mix thoroughly.
• Another sample is drawn, where we check for marked coins. This helps us understand how samples overlap.

By capturing and recapturing, we can apply a mathematical model to infer something about the whole. Through this exercise, we're given formulas to estimate the numbers of each type of coin, making it much easier to handle large quantities without exhaustive counting.

Mathematical Modeling

In situations where direct counting isn't feasible, mathematical modeling is employed to make educated estimates. Here, in the context of coin estimation, we use a mathematical formula that leverages the relationship between tagged and recaught items.The fundamental formula used is:\[ \text{Estimated Total} = \frac{\text{Number of first sample} \times \text{Number of second sample}}{\text{Number of recaptured tags}} \]This formula helps model the scenario and gives us the means to extrapolate from our sample back to the entire population. Here's the breakdown:

• The product of the two samples approximates the effect of total items interacting.
• The division by recaptured tags compares overlap, giving us a ratio reflective of the entire jar.

By applying this model separately to each type of coin, we can piece together an overall estimate, showcasing how mathematical modeling simplifies complex problems. This remarkable ability to infer large-scale data from smaller samples is just one example of mathematical modeling's power.

Estimation Techniques

Estimation techniques, like the one used here, help us solve problems efficiently without exhaustive efforts. The fundamental steps include sampling, labeling, and resampling, which form the backbone of the capture-recapture method. Here's a closer look:

• Sampling: Collect a representative subset of the whole.
• Labeling: Mark or tag your initial subset so they can be identified if encountered again.
• Resampling: Draw another sample and count how many of the labeled items are present.

These steps bridge the gap between impossible counts and usable estimates. By multiplying the sizes of the samples and dividing by the number of recaptured tags, we form reliable estimates.
The use of these techniques allows us not only to estimate in controlled situations, such as coin jars but also extends to more varied fields like wildlife management and quality control. Thus, we see that estimation is more than mere guesswork; it's a structured approach to gaining insight into larger populations.