Problem 27 - Algebra 2

Question

An event occurs \(x\) times in a sample of size \(n .\) Find its sample proportion and margin of error. $$ \begin{array}{l}{x=20} \\ {n=64}\end{array} $$

Step-by-Step Solution

Answer

The sample proportion is \(0.3125\) and the margin of error is approximately \(0.055\).

Step 1: Calculate Sample Proportion

Sample proportion 'p' is calculated using the formula \(p = x/n\). Here, \(x = 20\) and \(n = 64\). Plug these values into the formula to compute 'p' which equals \(20/64 = 0.3125\).

Step 2: Calculate Margin of Error

Now use the sample proportion 'p' to calculate the margin of error using the formula \(\sqrt{p*(1-p)/n}\). After substituting \(p = 0.3125\) and \(n = 64\), the formula becomes \(\sqrt{0.3125*(1-0.3125)/64}\), which equals approximately 0.055.

Key Concepts

Understanding Sample Proportion

The sample proportion is a way to express a subset of a total group as a fraction. In statistics, it helps us understand the relative frequency of an event happening within a given sample. The formula to calculate the sample proportion, denoted as \( p \), is simple and straightforward: it's the ratio of the number of successful events \( x \) to the total number of observations \( n \), expressed as \( p = \frac{x}{n} \).

In our exercise, where \( x = 20 \) and \( n = 64 \), the sample proportion \( p \) is computed as \( \frac{20}{64} \). Doing the math, you will find that \( p = 0.3125 \). This means that in our sample, about 31.25% of the observations were successful events.

Knowing the sample proportion gives us insight into the behavior of the dataset and provides a foundation for further statistical calculations. It tells us the "weight" or "importance" of a particular outcome in our sample.

Demystifying the Margin of Error

The margin of error is a crucial concept in statistics. It reflects the degree of uncertainty or potential error involved when estimating the characteristics of a population from a sample. The margin of error gives us an idea of how much the sample results are expected to vary from the true population value. Calculating it helps to build a confidence interval around the sample proportion, which indicates a range in which the actual population proportion is likely to fall.

To calculate the margin of error for a sample proportion, you use the formula: \[ \text{Margin of Error} = \sqrt{\frac{p(1-p)}{n}} \]Following the example from the exercise, where the sample proportion \( p = 0.3125 \) and the sample size \( n = 64 \), plug these values into the formula to find:

• First, calculate \( 1-p = 1 - 0.3125 = 0.6875 \)
• Then apply these into the formula: \( \sqrt{\frac{0.3125 \times 0.6875}{64}} \)

This gives a margin of error of approximately 0.055. This means that our sample proportion might differ from the true population proportion by roughly 5.5%, up or down.

Using Statistical Formulas to Simplify Problems

In probability and statistics, formulas are powerful tools. They allow us to systematically approach and solve problems. Each formula is derived from statistical theory and serves a specific purpose. In this problem, two important formulas are employed: the formula for the sample proportion \( p = \frac{x}{n} \) and the formula for the margin of error \( \sqrt{\frac{p(1-p)}{n}} \).

These formulas require inputs like the number of successes \( x \), the sample size \( n \), and the sample proportion \( p \). Once you have these values, plugging them into the formulas is usually straightforward. Each formula gives you a reliable statistical measure that can help interpret the data. For example:

• The sample proportion helps us quantify the ratio of successful events in the sample.
• The margin of error aids in establishing confidence intervals, which suggest a range for where the population parameter might lie.

By breaking down the problem into steps and using these formulas, complex statistical questions become easier and more manageable. In practice, these formulas are crucial for anyone working with data to ensure accuracy and reliability in their predictions and decisions.