Problem 17 - Algebra 2

Question

For Exercises \(12-21,\) find the margin of sampling error to the nearest percent. $$ p=67 \%, n=1500 $$

Step-by-Step Solution

Answer

The margin of sampling error is 2\%.

Step 1: Understand the Formula for Margin of Sampling Error

To find the margin of sampling error, use the formula for the standard error of a proportion: \( SE = \sqrt{\frac{p(1-p)}{n}} \). The margin of error at a 95% confidence level is approximately \( 1.96 \times SE \), which is the Z score for a 95% confidence level.

Step 2: Convert Percentage to Proportion

Convert the percentage \( p \) to a proportion by dividing by 100. Here, \( p = 67\% \), so the proportion is \( p = 0.67 \).

Step 3: Calculate the Standard Error (SE)

Substitute \( p = 0.67 \) and \( n = 1500 \) into the standard error formula: \( SE = \sqrt{\frac{0.67 \times (1-0.67)}{1500}} \). Simplify inside the square root first.

Step 4: Perform the Calculation

First, calculate \( 0.67 \times (1-0.67) = 0.67 \times 0.33 = 0.2211 \). Then, find \( \frac{0.2211}{1500} = 0.0001474 \). Finally, take the square root to get \( SE = \sqrt{0.0001474} \approx 0.01214 \).

Step 5: Calculate the Margin of Sampling Error

Multiply the standard error by 1.96 to find the margin of error: \( Margin = 1.96 \times 0.01214 \approx 0.0238 \).

Step 6: Convert to Percentage and Round

Convert the margin back to a percentage by multiplying by 100: \( 0.0238 \times 100 = 2.38 \% \). Round to the nearest whole percent: 2\%.

Key Concepts

Standard Error

The standard error is a measure that tells us how much variability we can expect in our sample statistic from sample to sample. It is especially used when we're dealing with proportions.

When calculating the standard error of a proportion, we use the formula:

• \( SE = \sqrt{\frac{p(1-p)}{n}} \)

Here, \( p \) is the sample proportion, and \( n \) is the sample size. The square root in the formula represents the expected spread of the sample statistic. In simple terms, the standard error helps us understand how much the sample proportion might differ if we were to take several different samples.

Understanding this concept is crucial because, to calculate the margin of sampling error, we first need to know what the standard error is. This will then allow us to determine the range within which the true proportion is expected to lie.

Confidence Level

Confidence level is the probability that the true population parameter will fall within the calculated range or interval. It is an expression of how sure we are that the range contains the true value.

The choice of confidence level affects the multiplier we use in calculating the margin of error. The most common confidence level is 95%. This implies a probability of 95% that the interval created will contain the actual population parameter.

In a normal distribution, the number associated with a 95% confidence level is 1.96. Therefore, to find the margin of error, we multiply the standard error by 1.96. The concept of a confidence level ensures that we're accounting for the natural variability in sampling and gives a quantifiable measure of certainty to our estimates.

Proportion

Proportion represents a part of the whole and is often expressed as a decimal or percentage. In statistics, we often deal with proportions when trying to understand what fraction of the total our data point represents.

In the given exercise, the proportion is derived from converting the percentage of 67% into a decimal: 0.67. This makes it easier for calculations. Knowing the proportion ahead of time is essential since it feeds directly into the standard error calculation. The greater the proportion, the more certain we can be about our estimates.

The proportion factor in the equation \( p(1-p) \) plays a key role by accounting for both the "successes" \( p \) and "failures" \( 1-p \) simultaneously. This dual calculation helps stabilize the mathematical model by considering all outcomes.

Sample Size

Sample size, denoted as \( n \), refers to the number of observations included in the sample. The size of the sample greatly affects both the standard error and the confidence in the results.

A larger sample size typically leads to a smaller standard error, which means more precise estimates. In our exercise, a sample size of 1500 makes the calculations more robust and the margin of error smaller than it would be with a smaller sample size.

Sample size is a critical consideration in many research designs as it determines the extent to which findings can be generalized to the broader population. By increasing the sample size, we reduce variability and the potential for sampling error, making it easier to trust the accuracy of our findings.