Problem 8 - Calculus for Biology and Medicine

Question

The following data represent the relative frequency distribution of clutch size in a sample of 42 mallards: $$ \begin{array}{cc} \hline \text { Clutch Size } & \text { Relative Frequency } \\ \hline 6 & 0.10 \\ 7 & 0.24 \\ 8 & 0.29 \\ 9 & 0.21 \\ 10 & 0.16 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

Step-by-Step Solution

Answer

Mean: 8; Variance: 1.49.

Step 1: Understand the Problem

We are given a relative frequency distribution of the clutch sizes of mallards. We need to calculate the sample mean and the sample variance from this distribution. The sample size is 42 mallards.

Step 2: Calculate Individual Frequencies

First, calculate the frequency for each clutch size by multiplying the relative frequency by the total sample size (42). For example, for clutch size 6: 0.10 * 42 = 4.2. Do this for each clutch size.

Step 3: Find the Sample Mean

The sample mean is calculated as the sum of the products of each clutch size and its frequency, divided by the total number of observations. Use the formula: \[ \bar{x} = \frac{\sum (x_i \times f_i)}{n} \]where \(x_i\) is the clutch size and \(f_i\) is the frequency of \(x_i\).

Step 4: Compute Clutch Size Frequencies

Calculate each clutch size frequency:\[ 6: 0.10 \times 42 = 4.2 \] \[ 7: 0.24 \times 42 = 10.08 \] \[ 8: 0.29 \times 42 = 12.18 \] \[ 9: 0.21 \times 42 = 8.82 \] \[ 10: 0.16 \times 42 = 6.72 \]

Step 5: Compute the Sample Mean

Calculate the sample mean using the clutch size frequencies and values:\[ \bar{x} = \frac{6\times 4.2 + 7\times 10.08 + 8\times 12.18 + 9\times 8.82 + 10\times 6.72}{42} \]\[ \bar{x} = \frac{6(4.2) + 7(10.08) + 8(12.18) + 9(8.82) + 10(6.72)}{42} \]\[ \bar{x} = \frac{6(4.2) + 7(10.08) + 8(12.18) + 9(8.82) + 10(6.72)}{42} = 8 \]

Step 6: Find the Sample Variance

The sample variance \(s^2\) is calculated using:\[ s^2 = \frac{\sum (x_i - \bar{x})^2 \times f_i}{n} \]Compute the squared deviations multiplied by their frequencies and divide by the number of observations.

Step 7: Calculate Squared Deviations

Calculate each squared deviation:\[ (6-8)^2 = 4 \] \[ (7-8)^2 = 1 \] \[ (8-8)^2 = 0 \] \[ (9-8)^2 = 1 \] \[ (10-8)^2 = 4 \]

Step 8: Compute Variance Products

Calculate each product of squared deviation and frequency:\[ 4 \times 4.2= 16.8 \] \[ 1 \times 10.08 = 10.08 \] \[ 0 \times 12.18 = 0 \] \[ 1 \times 8.82 = 8.82 \] \[ 4 \times 6.72 = 26.88 \]

Step 9: Calculate the Sample Variance

Add all variance products and divide by total sample size to find the variance:\[ s^2 = \frac{16.8 + 10.08 + 0 + 8.82 + 26.88}{42} \]\[ s^2 = \frac{62.58}{42} = 1.49 \]

Key Concepts

Relative Frequency Distribution

A relative frequency distribution provides information on how data points are spread across various categories. In this case, it shows how often different clutch sizes occur in a sample of mallard ducks relative to the entire sample.

Relative frequency is the proportion of the total number of observations that fall within a particular category. It is calculated by dividing the frequency of a particular outcome by the total sample size. The formula is:
\[ \text{Relative Frequency} = \frac{\text{Frequency of a category}}{\text{Total sample size}} \]

• For clutch size 6, the relative frequency is 0.10, meaning 10% of the ducks have this clutch size.
• Similarly, the clutch size 7 has a relative frequency of 0.24, which represents 24% of the sample.

This distribution helps us to quickly see which clutch sizes are most common and which are less frequent in the sample.

Sample Mean

The sample mean provides an average value for our data set, offering a quick summary measure of a group of data points. To find the sample mean of clutch sizes, we multiply each clutch size by its frequency, sum these results, and divide by the total number of data points in the sample.

The formula for the sample mean is:\[ \bar{x} = \frac{\sum (x_i \times f_i)}{n} \]

• In our example, the sample mean is calculated using clutch sizes such as 6, 7, and 8...
• ...and their frequencies from the distribution table.

' This summary statistic, the mean, offers a central tendency insight, indicating that 8 is the average clutch size for the mallards in this sample.

Frequency Calculation

Frequency calculation involves determining how often each clutch size appears within our sample. We do this by multiplying the relative frequency by the total sample size. This method provides an estimate of actual numbers when dealing with percentages.

The formula for calculating each category's frequency is straightforward:
\[ \text{Frequency} = \text{Relative Frequency} \times \text{Total Sample Size} \]

• For a clutch size of 6, we multiply 0.10 by 42 to get 4.2.
• Repeat this calculation for each clutch size to get all frequencies.

This calculation is important as it translates the relative frequencies into actual counts, making it easier to understand the number of occurrences of each clutch size.

Clutch Size

Clutch size refers to the number of eggs laid in a single nesting attempt by a bird, such as a mallard. It can provide vital information about the reproductive success and behavior of a species. In our example, we are studying different sizes of egg clutches in mallards, ranging from 6 to 10 eggs.

Understanding clutch sizes can help researchers:

• Monitor population growth or decline.
• Study changes in reproductive behavior over time.
• Assess impacts of environmental changes on breeding patterns.

By examining the distribution of clutch sizes, we can gain insights into the ecological and biological factors affecting species reproduction.