Question

A sample of 20 frogs has a total weight of \(W\) grams. (a) What is the mean weight (in grams per frog) of the sample? (b) One of the frogs has been mis-weighed. Instead of \(x\) grams, its weight is \(y\) grams. What is the corrected mean weight of the sample?

Short answer

Question: In a sample of 20 frogs, the total weight was found to be W grams. After discovering that one frog's weight was mis-weighed as x grams instead of the correct y grams, calculate the mean weight of the sample for both the uncorrected and corrected weights. Answer: The uncorrected mean weight of the sample is \(\frac{W}{20}\) grams per frog. The corrected mean weight is \(\frac{W - x + y}{20}\) grams per frog.

Step-by-step solution

Calculate the mean weight of the sample

We have a sample of 20 frogs with a total weight of W grams. The mean weight is calculated by dividing the total weight by the number of frogs: Mean weight = \(\frac{W}{20}\)

Find the initial sum of frog weights

Before we determine the corrected mean weight, let's first find the sum of all frog weights before the mis-weighing is corrected: Initial sum of weights = \(W\)

Calculate the corrected sum of frog weights

We know that one of the frog's weight was mis-weighed and was recorded as x grams instead of the correct y grams. We need to remove the mis-weighed x grams from the sum and replace it with the correct y grams: Corrected sum of weights = \(W - x + y\)

Calculate the corrected mean weight of the sample

Now that we have the corrected sum of frog weights, we can determine the corrected mean weight. We'll divide the corrected sum of weights by the number of frogs (20) to obtain the corrected mean weight: Corrected mean weight = \(\frac{W - x + y}{20}\)

Key concepts

Understanding Samples

The term 'sample' refers to a subset of a larger group, known as the population. In this case, the sample consists of 20 frogs taken from a potentially larger group of frogs. When we want to learn something about a population, we can often learn a lot by studying just a sample if the sample represents the larger population accurately. For example, if this sample of 20 frogs is chosen randomly, it can give us a good idea of what we might see in the entire population of frogs. In statistics, analyzing a sample helps make inferences about the population without examining every single member.

Applying Corrections to Data

Mistakes in data can often happen; for example, a mis-weighed frog in our study. When errors like these occur, they need to be corrected to avoid skewed results. Here, the process involves identifying the error (the frog recorded as weighing \(x\) grams instead of the correct \(y\) grams) and applying a correction.To make a correction, you adjust the total recorded weight of the sample:

• Subtract the incorrect weight (\(x\)) from the total sum.
• Add the correct weight (\(y\)) to the sum.

This gives us the corrected sum of weights.These corrections help maintain the accuracy and reliability of our data, ensuring that any conclusions drawn from the data are valid.

Performing Calculations for Mean Weight

Calculations are at the heart of determining the mean weight of a sample. The mean weight tells us the average weight per frog.To find the mean, you divide the total weight of the frogs by the number of frogs: \[ \text{Mean weight} = \frac{W}{20} \]This gives us the initial average weight per frog. When corrections are necessary, we adjust our total weights as discussed and perform the calculation again to get the corrected mean: \[ \text{Corrected mean weight} = \frac{W - x + y}{20} \]These calculations are essential for ensuring that our data accurately reflects the real-world situation, allowing informed conclusions to be made based on this analysis.