Question

Find the sample proportion \(\hat{p}\). The survey students consisted of 169 females and 193 males. Find \(\hat{p},\) the proportion who are female.

Short answer

The proportion of students who are female is 0.467 or 46.7%.

Step-by-step solution

Identify the 'success' and total sample size

In this case, we can think of 'success' as being a female, so \(x = 169\). The total number of students (the sample size) is \(n = 169+193=362\).

Apply the formula

Substitute the numbers into the formula \(\hat{p} = \frac{x}{n}\), which gives us \(\hat{p} = \frac{169}{362}\)

Compute the sample proportion

By doing the above calculation, we find that \(\hat{p}\) equals approximately 0.467. Therefore, the proportion of students who are female is about 0.467 or 46.7%.

Key concepts

Statistical Sampling

Statistical sampling is a powerful tool that allows researchers to make inferences about a larger population based on a subset of that population, known as a sample. Imagine you have a large vat of delicious soup. Rather than tasting the entire vat to check for the right seasoning, you take a spoonful – that's your sample. In the case of survey data, like the one where we're determining the proportion of female students, we're effectively 'tasting' the characteristics of a larger population (the entire student body) by examining a smaller group (the 362 surveyed students).

When creating a sample, it is crucial to ensure that it is representative of the population, which means it needs to be randomly selected to avoid bias. In our example, if the sample of students surveyed is randomly selected, the proportion of female students in the sample should provide a good estimate of the proportion in the full student body. On the other hand, if the sample is not random, this could lead to an incorrect estimation, such as overestimating or underestimating the proportion of female students. Therefore, statistical sampling is not just about numbers; it's about the process of choosing those numbers to accurately reflect the whole.

Survey Data Analysis

Survey data analysis is a pivotal step in understanding what your data is telling you, particularly when you're looking at characteristics like gender, opinions, or behaviors within a population. Once you've gathered survey data using good sampling techniques, the real work begins: making sense of that data.

To do this effectively, one must be able to organize, summarize, and interpret the data. For instance, in the exercise, we aren't just counting students; we are specifically analyzing the count of female students relative to the total. This kind of analysis helps to draw meaningful conclusions from the survey, like understanding gender distribution within a student population. Analyzing the survey data requires careful attention to detail and knowledge of the right statistical tools in order to avoid misinterpretations that could lead to wrong conclusions or decisions based on the survey's findings.

Proportion Calculation

Proportion calculation is a mathematically straightforward yet informative statistical technique used to determine the relative frequency of a particular outcome within a dataset. Think of it as finding out what slice of the pie you're looking at in a pie chart. In the student survey example, the proportion of female students is calculated to understand their representation within the surveyed group.

To calculate a sample proportion, you divide the number of 'successes' (in our example, the number of females, which is 169) by the total number of observations in the sample (362). The formula looks like this: \(\hat{p} = \frac{x}{n}\), where \(x\) is the number of successes and \(n\) the sample size. This proportion, \(\hat{p}\), can then be expressed as a decimal, like 0.467, or a percentage, 46.7%. Understanding how to calculate proportions is fundamental in analyzing survey results, as it provides a quick means to quantify how a part relates to the whole, allowing for comparisons and further statistical analysis.