Problem 12 - Algebra 2

Question

For Exercises \(12-21,\) find the margin of sampling error to the nearest percent. $$ p=81 \%, n=100 $$

Step-by-Step Solution

Answer

The margin of error is approximately 8%.

Step 1: Understand the Formula for Margin of Error

The margin of error can be calculated using the formula: \( \text{ME} = Z \sqrt{\frac{p(1-p)}{n}} \), where \( Z \) is the z-score corresponding to the desired confidence level, \( p \) is the sample proportion, and \( n \) is the sample size.

Step 2: Convert Percentage to Decimal

Convert the percentage \( p = 81\% \) to a decimal: \( p = 0.81 \). This is necessary for use in the formula.

Step 3: Determine Z-score for Desired Confidence Level

Assume a common confidence level, such as 95%. The corresponding z-score is approximately \( 1.96 \).

Step 4: Plug Values into Formula

Substitute \( p = 0.81 \), \( n = 100 \), and \( Z = 1.96 \) into the margin of error formula: \[ ME = 1.96 \times \sqrt{\frac{0.81 \times (1-0.81)}{100}} \]

Step 5: Compute the Margin of Error

Calculate the expression inside the square root first: \( \frac{0.81 \times 0.19}{100} = 0.001539 \).Then take the square root: \( \sqrt{0.001539} \approx 0.0392 \).Finally, multiply by 1.96: \( ME \approx 1.96 \times 0.0392 \approx 0.0768 \).

Step 6: Convert the Margin of Error to Percentage

Multiply the margin of error by 100 to convert it from a decimal to a percentage: \( 0.0768 \times 100 \approx 7.68\% \).

Step 7: Round the Margin of Error to Nearest Whole Percent

Round 7.68% to the nearest whole number: The margin of error is approximately 8%.

Key Concepts

Sampling Error

Sampling error is the difference between the sample results and the true population results. It arises because a sample, rather than the whole population, is being analyzed. This can lead to slight variations in results from what is true for the entire population.
This error is unavoidable but can be minimized by using a larger sample size and employing a well-conducted random sampling method.

• The larger the sample size, the smaller the sampling error.
• Random sampling helps in minimizing biases, ensuring that every member of the population has an equal chance of being included in the sample.

Understanding sampling error is crucial because it informs us about the potential deviation we might expect in our sample result from the actual population parameter.

Confidence Level

The confidence level reflects how certain we can be that our results are reliable and thus close to real population values. It is often expressed as a percentage, like 95%, indicating that if we were to take many samples, we expect the true parameter to fall within our margin of error 95% of the time.

• A higher confidence level requires a broader margin of error.
• Typically, common confidence levels are 90%, 95%, and 99%.

Choosing the appropriate confidence level depends on how precise you need your results to be and the total amount of error you are willing to tolerate in the context of your study.

Z-score

The Z-score is a statistical measurement that describes a value's position relative to the mean of a set of values, measured in terms of standard deviations. In the context of confidence levels, the Z-score corresponds to a specific confidence level.

• For a 95% confidence level, the Z-score is approximately 1.96.
• The Z-score helps determine how far and in what direction a data point is from the mean.

When calculating the margin of error, the Z-score is used to quantify the variability that one might expect based on the chosen confidence level.

Sample Proportion

The sample proportion, represented as \( p \), is an estimate of the population proportion and is calculated by dividing the number of successful occurrences by the total number of trials. In our exercise, it was given as 81%, or 0.81 in decimal form for calculations.

• The sample proportion is critical in determining the margin of error as it provides a snapshot of how the sample represents the larger population.
• It is used in the formula to reflect the proportion of a particular attribute within the sampled data.

Without the sample proportion, it would not be feasible to estimate how well your sample statistics approximate the corresponding population parameters.

Sample Size

Sample size \( n \), is the number of observations in the sample. It plays a crucial role in any statistical analysis as it affects the accuracy and reliability of the results.

• Larger samples tend to yield more reliable estimates, reducing the margin of error.
• A small sample might not represent the population adequately, increasing the sampling error.

The sample size is a key component in the formula for the margin of error, as it inversely affects the margin, meaning as sample size increases, the margin of error decreases. Understanding its impact can help in designing more effective surveys and studies.