Problem 3 - Algebra 2

Question

For Exercises \(3-5,\) find the margin of sampling error to the nearest percent. $$ p=72 \%, n=100 $$

Step-by-Step Solution

Answer

The margin of sampling error is approximately 9%.

Step 1: Understand the Formula

To find the margin of sampling error for a proportion, we use the formula \( ME = z \times \sqrt{\frac{p(1-p)}{n}} \), where \( z \) is the z-score (usually 1.96 for 95% confidence), \( p \) is the proportion (in decimal form), and \( n \) is the sample size.

Step 2: Convert Percentage to Decimal

Convert the percentage \( p = 72\% \) to a decimal for calculation. This is done by dividing by 100. Thus, \( p = 0.72 \).

Step 3: Calculate Standard Error

Calculate the standard error using the formula \( \sqrt{\frac{p(1-p)}{n}} \). Substitute \( p = 0.72 \), \( 1-p = 0.28 \), and \( n = 100 \): \[ \sqrt{\frac{0.72 \times 0.28}{100}} = \sqrt{\frac{0.2016}{100}} = \sqrt{0.002016} \approx 0.0449 \]

Step 4: Apply the Z-score

Multiply the standard error by the z-score for a 95% confidence level, which is 1.96. \[ ME = 1.96 \times 0.0449 \approx 0.088 \]

Step 5: Convert Margin of Error to Percentage

Convert the margin of error back to a percentage by multiplying by 100. Therefore, \( ME = 0.088 \times 100 \approx 8.8\% \).

Step 6: Round to Nearest Percent

Round the margin of error to the nearest whole percentage. Hence, the margin of error is approximately \( 9\% \).

Key Concepts

z-score

The z-score is a statistical measurement that describes the relationship of a data point to the mean of a group of data points. Essentially, it measures how many standard deviations an element is from the mean. In the context of margin of error calculations,

• the z-score is often used to express the result within a confidence interval.
• It serves as a multiplier in our formula: \( ME = z \times \text{Standard Error} \), determining the margin of error for a desired level of confidence.
• For a 95% confidence interval, the z-score is typically 1.96.

This means that approximately 95% of data is expected to fall within 1.96 standard deviations from the mean in a normal distribution. Thus, selecting an appropriate z-score is crucial to accurately estimate the range in which the actual population parameter would lie based on your sample.

standard error

The standard error represents how much sample means are expected to deviate from the true population mean. When estimating a proportion, we calculate the standard error using the formula \[\text{Standard Error} = \sqrt{\frac{p(1-p)}{n}} \]where

• \(p\) is the sample proportion
• \(1-p\) represents the proportion not in the sample
• \(n\) is the sample size

The concept of standard error is critical in understanding variability in sample estimates. Essentially,

• a smaller standard error indicates that the sample mean is closer to the population mean, suggesting more precise estimates.
• Larger sample sizes will generally lead to smaller standard error, increasing the confidence in the sample mean as a good estimator of the population mean.

confidence interval

A confidence interval provides a range within which we expect the true parameter of the population to lie, based on the sample data. This range gives us an interval with a specified probability of containing the true value, with the confidence level representing this probability.

• The confidence interval is calculated as the sample statistic plus and minus the margin of error: \( \text{CI} = \hat{p} \pm \text{ME} \).
• Here, \( \hat{p} \) represents the sample proportion and \( \text{ME} \) the margin of error.
• If we set a 95% confidence interval, we are saying there is a 95% chance that the interval contains the true population proportion.

By using a confidence interval, researchers can better understand the range and certainty of their estimation, ensuring important decisions are based on statistical evidence.

proportion

A proportion in statistics represents the size of a subset relative to its larger set. It is typically expressed as a fraction or a percentage. In sampling and estimation,

• proportions allow us to measure the characteristics of a sample compared to the entire population.
• For example, if we know that 72% (or 0.72 as a fraction) of a sample prefers a certain product, this proportion helps in estimating the preference in the whole population.
• The role of proportion in the margin of error formula is significant as it implicates sample variance: \[p \times (1 - p)\]where a greater balance between \( p \) and \( (1-p) \) implies more uncertainty, reflected in a larger standard error.

Understanding proportion is fundamental in various fields like market research, polling, and any domain where predicting population features from samples is necessary.