Problem 7 - Calculus for Biology and Medicine

Question

The following data represent the relative frequency distribution of clutch size in a sample of 300 laboratory guinea pigs: $$ \begin{array}{cc} \hline \text { Clutch Size } & \text { Relative Frequency } \\ \hline 2 & 0.05 \\ 3 & 0.09 \\ 4 & 0.12 \\ 5 & 0.19 \\ 6 & 0.23 \\ 7 & 0.12 \\ 8 & 0.13 \\ 9 & 0.07 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

Step-by-Step Solution

Answer

Sample Mean is approximately 5.78, and Sample Variance is approximately 2.73.

Step 1: Identify the Values Needed

First, recognize that you will use the clutch size as your variable and the relative frequency will be used to compute both the mean and the variance. The relative frequency must be multiplied by the total sample size to convert it into actual frequency.

Step 2: Calculate the Actual Frequencies

For each clutch size, multiply the relative frequency by the total number of guinea pigs (300) to find the actual frequencies. For instance, the frequency for clutch size 2 is calculated as \( 0.05 \times 300 = 15 \). Repeat this for all clutch sizes.

Step 3: Compute the Sample Mean

The sample mean \( \bar{x} \) is calculated by multiplying each clutch size by its frequency, summing all these products, and then dividing by the total number of observations \( n \). That is, \( \bar{x} = \frac{\sum (x_i \times f_i)}{n} \), where \( x_i \) are clutch sizes and \( f_i \) are their respective frequencies.

Step 4: Calculate the Deviation from the Mean

For each clutch size, find the square of the difference between the clutch size and the mean. Multiply each squared deviation by its frequency to get the weighted squared deviation.

Step 5: Compute the Sample Variance

The sample variance \( s^2 \) is calculated using the formula \( s^2 = \frac{\sum f_i (x_i - \bar{x})^2}{n-1} \). Divide the total weighted squared deviation from Step 4 by \( n-1 \) to find the variance.

Key Concepts

Relative Frequency Distribution

Relative frequency distribution is a way to display how often an event happens in a dataset. Instead of giving the count of each occurrence, it shows the proportion of the dataset that each item represents. This is done by dividing the frequency of a particular event by the total number of observations.

For example, in a sample of 300 guinea pigs, if we want to know the proportion of guinea pigs with a clutch size of 2, we look at the relative frequency given as 0.05. This indicates that 5% of the guinea pigs have a clutch size of 2.

• Example: For clutch size 6 with a relative frequency of 0.23, 23% of guinea pigs have this clutch size.
• Sum of all relative frequencies should equal 1.

The concept helps us quickly understand the distribution pattern of outcomes without needing to calculate actual counts. Simply multiply the relative frequency by the total sample to get actual counts. This can be especially useful when working with large datasets or when approximations are sufficient.

Sample Mean

The sample mean is a measure that provides the average value of a dataset. It is calculated by taking the sum of all data values and then dividing by the total number of observations.

For the guinea pigs' clutch sizes, the sample mean helps us understand the average clutch size in the sample of guinea pigs. First, multiply each clutch size by its respective frequency to get the total sum for all observations. Then divide this sum by the total sample size to find the mean.

The formula for calculating the sample mean \( \bar{x} \) is:

\[\bar{x} = \frac{\sum (x_i \times f_i)}{n}\]
where \( x_i \) represents each clutch size, \( f_i \) is the frequency of each clutch size, and \( n \) is the total number of data points (300 in this case).

Calculating the mean gives a quick, concise summary of the central tendency, offering a single representative value of the dataset.

Clutch Size

Clutch size refers to the number of offspring a female, such as a guinea pig, produces at one time. This is an important biological characteristic that can vary among species and even between individuals of the same species. In the context of laboratory guinea pigs, clutch size can provide insights into reproductive trends, genetic health, and overall population dynamics.

Understanding clutch size can help in breeding programs and in drawing ecological or evolutionary conclusions. It plays a crucial role in population studies, affecting how researchers interpret data, create models, and develop strategies for animal management or conservation efforts.

In practical terms, scientists might analyze clutch size data to identify optimal breeding conditions or to understand how environmental factors influence reproduction in guinea pigs. By analyzing variations in clutch size, researchers can better understand both individual and population health.