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In the context of Mendel's genetics experiment, the sample proportion is crucial. It represents the proportion of green peas in our sample of 580 peas.

To calculate it, you simply divide the number of green peas by the total number of peas. In this exercise:
\[ \text{Sample Proportion} = \frac{428}{580} \]
This gives us approximately 0.7379. This value indicates that about 73.79% of the peas in this sample are green.

Understanding sample proportion helps us in estimating and making predictions about the population from which the sample is drawn.

The standard error is another critical concept. It measures the variability of our sample proportion. Specifically, it tells us how much the sample proportion (0.7379) would vary from sample to sample if we were to take many samples.

The formula for the standard error of a sample proportion is:
\[\text{SE} = \frac{ \text{Sample Proportion} \times (1 - \text{Sample Proportion})}{ N }\bigg]^{1/2}\]
Where:

• Sample Proportion (p) = 0.7379
• N = 580

Plugging in our values:
\[\text{SE} = \bigg[\frac{0.7379 \times (1 - 0.7379)}{580}\bigg]^{1/2} = 0.0186 \]
The smaller the standard error, the more precise our estimate of the population proportion.

The margin of error gives us a range within which we can be confident that the true population proportion lies.

To find the margin of error, we need the critical value (z*) for our desired confidence level. For a 99% confidence level, z* is approximately 2.576. The formula to compute the margin of error is:
\[\text{ME} = z^* \times \text{SE} \]
Given our standard error of 0.0186, the margin of error is:
\[\text{ME} = 2.576 \times 0.0186 = 0.0479 \]
Adding and subtracting this margin from our sample proportion gives us the confidence interval.

This interval helps us understand the precision of our estimate.

In this genetics experiment, Mendel expected 75% of the peas to be green based on his theory. With a 99% confidence interval calculated as (0.6900, 0.7858), we need to interpret this in the context of Mendel's theory.

Since the interval includes 75% (0.75), it means that the observed sample proportion (73.79%) does not significantly contradict his expected proportion. This gives us evidence that the results of this experiment support Mendel's theory.

Understanding the science behind Mendel's genetics and statistical concepts allows us to draw meaningful conclusions from experimental data. This reinforces the importance of combining sound experimental design with rigorous statistical analysis.