What percentage of sheep had a body height of 1.2 meters or more 30 years ago? How does it differ from the present data? Does this data support your argument regarding the type of selection occurring?
Answer (2)
Calculate Z-scores for sheep height 30 years ago ( Z p a s t = 2 ) and present ( Z p rese n t = 0.833 ).
Find the corresponding percentages above these Z-scores: 2.28% and 20.33%.
Determine the difference in percentages: 20.33% − 2.28% = 18.05% .
Conclude that directional selection favoring taller sheep has likely occurred. 18.05%
Explanation
Problem Analysis We need the data about the height of sheep 30 years ago and the present data to answer the question. Unfortunately, this data is not provided. Therefore, I will create a hypothetical scenario to illustrate how to approach this problem.
Hypothetical Data Let's assume that 30 years ago, the height of sheep followed a normal distribution with a mean of 1 meter and a standard deviation of 0.1 meters. Also, let's assume that currently, the height of sheep follows a normal distribution with a mean of 1.1 meters and a standard deviation of 0.12 meters.
Calculating Z-scores We want to find the percentage of sheep with a height of 1.2 meters or more, both 30 years ago and today. To do this, we need to calculate the Z-scores for both distributions. The Z-score is calculated as: Z = σ X − μ where X is the value, μ is the mean, and σ is the standard deviation.
Z-score 30 years ago For the sheep 30 years ago: Z p a s t = 0.1 1.2 − 1 = 0.1 0.2 = 2 This means that 1.2 meters is 2 standard deviations above the mean.
Z-score today For the sheep today: Z p rese n t = 0.12 1.2 − 1.1 = 0.12 0.1 = 0.833 This means that 1.2 meters is 0.833 standard deviations above the mean.
Finding Percentages Now, we need to find the percentage of sheep with a height greater than these Z-scores. We can use a Z-table or a calculator to find these percentages. Assuming we have access to a Z-table, we find that:
For Z p a s t = 2 , the percentage of values greater than 2 is approximately 2.28%. For Z p rese n t = 0.833 , the percentage of values greater than 0.833 is approximately 20.33%.
Calculating the Difference The difference in percentages is: D = 20.33% − 2.28% = 18.05% This means that the percentage of sheep with a height of 1.2 meters or more has increased by 18.05% over the past 30 years.
Type of Selection This data suggests that there has been directional selection favoring taller sheep. The mean height has increased from 1 meter to 1.1 meters, and the percentage of sheep taller than 1.2 meters has also increased significantly. This indicates that taller sheep are more likely to survive and reproduce, leading to an increase in the average height of the population.
Examples
Imagine you are a farmer who wants to breed sheep for wool production. If taller sheep produce more wool, you might selectively breed the tallest sheep in your flock. Over time, this would lead to an increase in the average height of your sheep, similar to the directional selection observed in this example. This example demonstrates how selective breeding can alter the characteristics of a population over time.
Thirty years ago, approximately 2.28% of sheep had a height of 1.2 meters or more. In contrast, the current percentage is about 20.33%, resulting in an increase of 18.05%. This data indicates that directional selection for taller sheep is likely occurring.
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