Problem 20 - Algebra Form and Function

Question

A sample of \(n_{1}\) frogs has a total weight of \(W_{1}\) grams. A second sample of \(n_{2}\) frogs has a total weight of \(W_{2}\) grams. (a) What is the mean weight (in grams per frog) of each sample? (b) What is the mean weight of all the frogs in both samples? Does it equal the average of the mean weights of the samples taken separately?

Step-by-Step Solution

Answer

Explain your reasoning. Answer: No, the mean weight of all frogs in both samples is not always equal to the average of the mean weights of the samples taken separately. This is because when we compare the two expressions and simplify the equation, it does not hold true for all cases.

Step 1: Find Mean Weight of Each Sample separately

To find the mean weight of each sample, we'll divide the total weight of the sample by the number of frogs in the sample. For Sample 1: Mean Weight = \(\frac{W_1}{n_1}\) For Sample 2: Mean Weight = \(\frac{W_2}{n_2}\)

Step 2: Find Mean Weight of All Frogs in Both Samples

To find the mean weight of all the frogs in both samples, we will first find the total number of frogs and the total weight of all the frogs in both samples. Total Number of Frogs = \(n_1 + n_2\) Total Weight = \(W_1 + W_2\) Mean Weight of All Frogs = \(\frac{Total Weight}{Total Number of Frogs}\) Mean Weight of All Frogs = \(\frac{W_1 + W_2}{n_1 + n_2}\)

Step 3: Check If Mean Weight of All Frogs Equals the Average of the Mean Weights of the Samples

To check if the mean weight of all frogs is the same as the average of the mean weights of the samples, we will calculate the average of the mean weights of the samples and compare the value with the mean weight of all the frogs. Average of Mean Weights of Samples = \(\frac{\frac{W_1}{n_1} + \frac{W_2}{n_2}}{2}\) Now let's simplify the average of the mean weights of the samples: \(= \frac{\frac{W_1n_2 + W_2n_1}{2n_1n_2}}\) If the mean weight of all frogs (\(\frac{W_1 + W_2}{n_1 + n_2}\)) is equal to the average of the mean weights of the samples (\(\frac{\frac{W_1n_2 + W_2n_1}{2n_1n_2}}\)), then the statement is true. Otherwise, it is false. Mean Weight of All Frogs = \(\frac{W_1 + W_2}{n_1 + n_2}\) Average of Mean Weights of Samples = \(\frac{\frac{W_1n_2 + W_2n_1}{2n_1n_2}}\) For both values to be equal, the following condition must be true: \(\frac{W_1 + W_2}{n_1 + n_2} = \frac{\frac{W_1n_2 + W_2n_1}{2n_1n_2}}\) Cross-multiply: \(2n_1n_2(W_1 + W_2) = (n_1 + n_2)(W_1n_2 + W_2n_1)\) Simplify: \(2n_1n_2W_1 + 2n_1n_2W_2 = n_1W_1n_2 + n_1W_2n_1 + n_2W_1n_2 + n_2W_2n_1\) Then, see if both sides are the same: \(2n_1n_2W_1 + 2n_1n_2W_2 = n_1W_1n_2 + n_1W_2n_1 + n_2W_1n_2 + n_2W_2n_1\) This equation does not simplify to be true in all cases, so the mean weight of all the frogs in both samples does not always equal the average of the mean weights of the samples taken separately.

Key Concepts

Mean Weight

When discussing the concept of mean weight in algebra, it's crucial to understand what it signifies. Mean weight refers to the average weight per unit of a group, in this case, frogs. We calculate this by dividing the total weight of the group by the number of units (frogs) present. This gives us a simple yet effective way to understand the weight distribution within a sample. To find the mean weight of each sample:

• For the first group of frogs: divide their total weight, denoted as \(W_1\), by the number of frogs, \(n_1\). This gives the mean as \( \frac{W_1}{n_1} \).
• Similarly, for the second group, the mean is \( \frac{W_2}{n_2} \).

These calculations will help us understand if one group, on average, weighs more than the other.

Sample Comparison

Comparing two samples typically involves calculating and analyzing their means. When we receive data from different groups, like two samples of frogs, we're interested in seeing how their average measurements differ or resemble one another. In this context, comparing the mean weights is a vital step.

• First, we determine the mean weight for each sample individually using the calculations above.
• Then, we calculate the mean weight for all frogs in both samples combined. This involves adding their total weights together and their total number of frogs together, yielding \( \frac{W_1 + W_2}{n_1 + n_2} \).

The comparison aspect lies in observing whether this overall mean aligns with the separate sample means. Frequently, the overall mean does not simply equate to the arithmetic average of the individual sample means, which invites further analysis into architectural sample differences.

Cross-Multiplication

Cross-multiplication is a fundamental algebraic technique that's widely applicable in comparing ratios or fractions. It simplifies the process of determining equality between fractions and thus is key in comparing the mean weights in this scenario. To test if the mean weight of all frogs equals the average of the separate mean weights, algebraic manipulation through cross-multiplication is employed.

• We set the equation for the mean weight of all frogs equal to the average of individual sample means.
• Cross-multiplication is used to eliminate the fractions, allowing us to analyze if both sides of the equation equate.

This results in:\[ 2n_1n_2(W_1 + W_2) = (n_1 + n_2)(W_1n_2 + W_2n_1) \]Upon simplification, if the two sides do not balance for all cases, it highlights that the combined mean weight doesn't equal the average of the two separate means, a conclusion essential in sample examination.