Problem 15 - Algebra 2

Question

For Exercises \(12-21,\) find the margin of sampling error to the nearest percent. $$ p=48 \%, n=1000 $$

Step-by-Step Solution

Answer

The margin of sampling error is 3%.

Step 1: Identify given values

We are given that the sample proportion \( p \) is 48\% or 0.48, and the sample size \( n \) is 1000.

Step 2: Determine the formula for the margin of error

The formula for the margin of error \( E \) in sampling proportion is \( E = z \times \sqrt{\frac{p(1-p)}{n}} \). For a common confidence level of 95\%, the \( z \)-value is typically 1.96.

Step 3: Calculate the standard error

Substitute \( p = 0.48 \) and \( n = 1000 \) into the standard error formula: \( \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.48 \times 0.52}{1000}} \approx 0.0158 \).

Step 4: Calculate the margin of error

Use the formula \( E = 1.96 \times 0.0158 \approx 0.0310 \). Convert this to a percentage by multiplying by 100, yielding \( 3.10\% \).

Step 5: Round the margin of error

Round \( 3.10\% \) to the nearest percent, which gives \( 3\% \).

Key Concepts

Sample Proportion

In statistics, the sample proportion is a crucial concept. It is a way to estimate the true proportion of a population based on a sample. Imagine you want to know what percentage of people in a town support a new policy. You can't ask everyone, so you survey a smaller group (the sample) instead. If 48% of the sampled group supports the policy, then the sample proportion is 0.48. This number gives you an idea of what the population feels, but there's always some uncertainty involved since it's based on a sample, not the whole group. Finding the margin of sampling error helps to understand how close the sample proportion might be to the true population proportion.

Confidence Level

The confidence level in statistics is all about certainty. It tells you how sure you can be about your sampling results reflecting the actual population. Commonly, the confidence level is set at 95%. This means if you were to take 100 different samples and calculate the interval estimate from each sample, 95 of these intervals are expected to contain the true population parameter. Choosing a higher confidence level means you're more certain, but it also means your margin of error will be wider. It's a balance between certainty and precision. A 95% confidence level is often used because it offers a good mix of both.

Z-value

In statistics, the z-value is related to the confidence level. It represents the number of standard deviations a data point is from the mean. When you calculate the margin of error for a proportion, the z-value helps control how wide your margin is. For a 95% confidence level, the z-value is usually about 1.96. This number is derived from standard normal distribution tables. Using this specific z-value ensures that there's a 95% chance the true population parameter is captured within the calculated range, assuring us of a high level of confidence with our sample findings.

Standard Error

The standard error is a key piece in estimating how much your sample proportion might differ from the true population proportion.It's calculated using the formula: \( \sqrt{\frac{p(1-p)}{n}} \), where \( p \) is the sample proportion, and \( n \) is the sample size. This formula gives you a measure of the variability or spread of the sample proportion.In simple terms, the standard error tells you how much you can expect your sample proportion to "bounce around" the true population proportion if you took multiple samples. A smaller standard error suggests more precise estimates, meaning your sample proportion is likely to be closer to the true value.