Question

In an experiment with a certain mutation in the fruitfly Drosophila, \(n\) individuals were examined; of these, \(20 \%\) were found to be mutants. Determine the standard error of \(\widetilde{P}\) if (a) \(n=100\) (20 mutants). (b) \(n=400\) (80 mutants).

Short answer

The standard error of \(\widetilde{P}\) when \(n=100\) is 0.04 and when \(n=400\) is 0.02.

Step-by-step solution

Understanding Standard Error of Proportion

The standard error of the sample proportion \(\widetilde{P}\) can be calculated using the formula \( SE = \sqrt{\frac{P(1 - P)}{n}} \), where \(P\) is the population proportion of interest and \(n\) is the sample size.

Calculate the Standard Error for n=100

For \(n=100\) with \(20\%\) mutants, the sample proportion \(\widetilde{P}\) is \(0.20\). Calculate the standard error using the formula \( SE = \sqrt{\frac{0.20(1 - 0.20)}{100}} = \sqrt{\frac{0.20 \times 0.80}{100}} = \sqrt{0.0016} = 0.04 \).

Calculate the Standard Error for n=400

For \(n=400\) with \(20\%\) mutants, the sample proportion \(\widetilde{P}\) is \(0.20\) again. Calculate the standard error using the formula \( SE = \sqrt{\frac{0.20(1 - 0.20)}{400}} = \sqrt{\frac{0.20 \times 0.80}{400}} = \sqrt{0.0004} = 0.02 \).

Key concepts

Understanding Sample Proportion

Sample proportion, often denoted as \(\tilde{P}\), is a statistic that estimates the fraction of individuals in a sample with a certain characteristic. For instance, in a study measuring the percentage of left-handed people in a classroom, if 3 out of 30 students are left-handed, the sample proportion of left-handers is \(\frac{3}{30} = 0.10\) or 10%.

It's a reflection of the population proportion, but it comes with variability because it's based on a subset of the population. The size of the sample and the variation within the population influence this variability. Larger sample sizes generally lead to sample proportions that better approximate the population proportion, which can be thought of as the true underlying probability of an individual having the characteristic of interest.

Population Proportion Explained

In contrast to the sample proportion, the population proportion is the actual proportion of individuals in the entire population that has a particular attribute. If we were to measure the mutation in every fruitfly of a species, rather than just a sample, we'd find the population proportion of that mutation.

Knowing the true population proportion is not always possible because it requires exhaustive data collection. Therefore, we use the sample proportion to estimate it. The difference between the sample and population proportions pertains to the concept of sampling error, which is inherent in the process of collecting data from only a subset of the entire group.

Sampling Distribution Essentials

The sampling distribution is a probability distribution that outlines how a given statistic, like the sample proportion, would behave if we repeated our sampling process multiple times. Imagine repeatedly taking samples of, say, 100 fruitflies, and calculating the proportion of mutants each time. The collection of these sample proportions creates the sampling distribution of the proportion.

It's a critical concept because it allows us to understand the probability associated with our sample results, and it's at the heart of inferential statistics. The central limit theorem suggests that this distribution will take on a normal shape as the sample size grows, providing the foundation for creating confidence intervals and hypothesis tests.

Statistical Inference Demystified

Statistical inference is a part of statistics that deals with making conclusions about a population based on data collected from a sample. It uses methods like hypothesis testing and confidence intervals to make educated guesses or inferences about population parameters, like the population proportion, based on sample statistics, like the sample proportion.

When we talk about the standard error of a sample proportion, we're referring to the standard deviation of its sampling distribution. It measures the typical distance that the sample proportion is from the population proportion. And as we've seen in the exercise, this standard error can be calculated, which then aids in crafting more precise inferences about the population as a whole.