Problem 29 - Algebra 2

Question

An event occurs \(x\) times in a sample of size \(n .\) Find its sample proportion and margin of error. $$ \begin{array}{l}{x=273} \\ {n=435}\end{array} $$

Step-by-Step Solution

Answer

The sample proportion is \(0.628\) and the margin of error is \(0.047\).

Step 1: Calculate the Sample Proportion

The sample proportion \(p\) is calculated as the ratio of the number of times the event occurs (\(x\)) to the total size of the sample (\(n\)). That is, \( p = \frac{x}{n} = \frac{273}{435} = 0.6275862068965517 \)

Step 2: Find the Margin of Error

The margin of error can be found using the formula: \( E = 1.96\sqrt{ \frac{p(1-p)}{n} } \). Substituting the values, we find \( E = 1.96\sqrt{ \frac{0.6275862068965517(1-0.6275862068965517)}{435} } = 0.04705666059755217 \)

Key Concepts

Margin of Error

The margin of error is a vital part of any survey or research. It represents the range within which we expect the true population parameter to fall, given our sample data. In simpler terms, if you repeated the study multiple times, you'd expect the true population value to fall within this range the majority of the time. The margin of error is directly influenced by the sample size and the variation in the data. For instance, in the step-by-step solution above, the margin of error is calculated as approximately 0.047 using a standard z-score of 1.96, which is typical for a 95% confidence level. This level of confidence is common in research to balance precision with practicality. A key takeaway is that the larger your sample size, the smaller the margin of error becomes. This is because larger samples tend to produce more reliable estimates of the population parameter.

Statistics

Statistics is the study of data and involves collecting, analyzing, interpreting, presenting, and organizing information. It's a crucial part of understanding and interpreting research data. In the context of the exercise, statistics was employed to calculate the sample proportion and the margin of error.

• The sample proportion gives us an estimate of the actual proportion of the population that has a given trait, based on our sample data.
• The margin of error offers insight into the precision of our estimate.

These components help in making informed predictions and decisions based on the data collected.

Probability

Probability plays a starring role in statistics, often serving as the foundation upon which many statistical theories are built. It is the measure of the likelihood that an event will occur, and it ranges from 0 (the event will not occur) to 1 (the event will occur). In the example provided, probability helps us understand the sample proportion. The sample proportion is essentially an estimate of the probability of observing the trait of interest in a larger population. In other words, if 0.628 or roughly 62.8% of our sample exhibits the trait, probability theory allows us to infer this same characteristic might appear in approximately 62.8% of the entire population, with our margin of error giving a sense of reliability.

Sample Size

Sample size refers to the number of observations in a sample, and it's a pivotal factor in statistical calculations. It directly affects the accuracy of the sample proportion and the margin of error. A larger sample size often leads to a more precise estimate of the population parameter, because it tends to mitigate the effects of random sampling error. For example, in the exercise, the sample size is 435. This reasonably sized sample helps ensure that the sample proportion is a good reflection of the true population proportion. Keep in mind:

• As sample size increases, the margin of error decreases.
• Larger sample sizes yield more reliable estimates but require more resources.

Choosing the appropriate sample size is crucial to the success of any statistical analysis.