Understanding the Proportions of a Normally Distributed Lemming Population

Lemmings are known for their natural body weight distribution, which can be modeled using a normal distribution. Given the mean body weight of these small rodents as 63.5 grams and a standard deviation of 12 grams, we can analyze specific proportions of this population using the Z-score formula and standard normal distribution tables.

Calculating the Proportions

We will calculate the proportion of the population that has a body weight of 78 grams and the proportion that is 78 grams or smaller. Let's break down the steps:

Step 1: Calculate the Z-score

The Z-score formula is given by:

Z (X - μ) / σ

Given:

Mean (μ) 63.5 grams Standard Deviation (σ) 12 grams Value of interest (X) 78 grams

Plugging in the values:

Z (78 - 63.5) / 12 14.5 / 12 ≈ 1.2083

Step 2: Find the Proportion Greater Than 78.0 grams

In a continuous distribution, the probability of any single point like exactly 78.0 grams is 0. Therefore, we will use the Z-score to determine the proportion of the population that is greater than 78.0 grams.

Step 3: Find the Proportion of the Population 78 grams or Smaller

To find the proportion of the population that is 78 grams or smaller, we need to find the cumulative probability up to the Z-score of 1.2083. Using the standard normal distribution table or a calculator, the cumulative probability for Z 1.2083 is approximately:

P(Z ≤ 1.21) ≈ 0.8869

Conclusion

Based on our calculations:

The proportion of the population that weighs exactly 78.0 grams is 0. The proportion of the population that weighs 78 grams or smaller is approximately 0.8869 or 88.69%.

Further Analysis

For a more precise analysis, let's consider the following scenarios:

Proportion of Population 78.0 grams

This proportion is not well defined due to the continuous nature of the distribution. However, we can calculate the proportion between two values.

Proportion of Population 78.0 grams or Smaller

If we are given a Z table:

Table that gives area from mean to Z: Table that gives area above Z (standard normal table): Table that gives area below Z (cumulative distribution table):

The key points to remember:

The total area under the bell curve is 1.0000. Half the area under the bell curve is 0.5000.

Using the cumulative distribution table, the proportion of the population that weighs 78 grams or smaller is:

Population ≤ 78.0 0.8869

This proportion is 88.69%.

Best of luck with your probability and statistics study! It is well worth learning the intricacies of normal distributions and Z-scores for real-world applications.