Problem 25 - Calculus for Biology and Medicine

Question

To determine the germination success of seeds of a certain plant, you plant 162 seeds. You find that 117 of the seeds germinate. Estimate the probability of germination and give a \(95 \%\) confidence interval.

Step-by-Step Solution

Answer

The probability of germination is approximately 0.7222 with a 95% confidence interval of (0.6519, 0.7925).

Step 1: Identify the Sample Proportion

First, we need to calculate the sample proportion (\(\hat{p}\)) of seeds that germinated. This is calculated by dividing the number of seeds that germinated by the total number of seeds planted. \[\hat{p} = \frac{117}{162} \approx 0.7222\]

Step 2: Calculate the Standard Error

Next, calculate the standard error (SE) of the sample proportion. The formula for standard error of a proportion is:\[SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\]Where \(n = 162\). Substitute \(\hat{p} = 0.7222\):\[SE = \sqrt{\frac{0.7222(1 - 0.7222)}{162}} \approx 0.0359\]

Step 3: Find the Z-Score for the Confidence Level

For a \(95\%\) confidence interval, we use a Z-score of approximately 1.96, which corresponds to the standard normal distribution's critical value for symmetric tails of \(2.5\%\) each (\(\frac{5\%}{2}\)).

Step 4: Calculate the Margin of Error

The margin of error (ME) can be calculated using the formula:\[ME = Z \times SE\]Substituting the given values, we have:\[ME = 1.96 \times 0.0359 \approx 0.0703\]

Step 5: Construct the Confidence Interval

Now, construct the \(95\%\) confidence interval using the sample proportion and the margin of error calculated:\[CI = \hat{p} \pm ME = 0.7222 \pm 0.0703\]Thus, the confidence interval is:\[(0.6519, 0.7925)\]

Key Concepts

Sample Proportion

The sample proportion is an estimate of the probability that a seed will germinate based on a small set of data, which, in this case, is the number of seeds planted and the number of seeds that actually germinated. This provides us with an idea of the true probability if we were able to plant an infinite number of seeds. In our specific example, we planted 162 seeds and 117 of those seeds germinated.
This scenario leads to a sample proportion (p) of 0.7222, calculated by dividing the number of germinating seeds by the total number of seeds.

This can be summarized as:

• Sample Proportion (p) = Number of Germinated Seeds / Total Number of Seeds
• p = 117 / 162  = 0.7222

This proportion serves as the foundation for further calculations, including the standard error and confidence intervals.
It helps to understand that the sample proportion is a point estimate, representing the probability based on this specific sample.

Confidence Interval

A confidence interval gives us a range within which we expect the true probability of success, such as the germination rate, to fall.
It provides more information than just the point estimate by showing us how precise our sample proportion is. With a confidence level of 95%, we are saying that we are 95% confident that the true germination rate falls within this interval.
In the provided solution, the confidence interval is calculated with the sample proportion (p = 0.7222) and the standard error.

The steps to calculate it include:

• Use the Z-score for your confidence level (for 95%, Z is about 1.96).
• Calculate the margin of error (ME) using the formula: ME = Z  SE.
• Add and subtract this margin of error from the sample proportion to find the interval.

Thus, the confidence interval derived here is (0.6519, 0.7925).
This means that, based on our sample, we expect the true probability of a seed germinating to lie between roughly 65% and 79%.

Standard Error

The standard error gives us a measure of the variability or spread of our sample proportion. It's essentially an estimation of how much the sample proportion (p) would vary if we repeated the sampling process many times under the same conditions.

Here's how you calculate it:

• Start with the formula for standard error (SE) of a proportion: \(SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
• Plug in your sample proportion (\(\hat{p}\)) and total number of observations (\(n\)).

In our example, p = 0.7222 and \(n\) = 162, leading to an SE of approximately 0.0359.
Understanding the standard error is key because it affects how wide or narrow our confidence interval becomes.

A smaller standard error results in a more precise/confident range, while a larger one suggests more uncertainty.
It's an essential aspect of estimating the reliability of our sample proportion.