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Magazine Chapter 12: Problem 3
For Exercises \(3-5,\) find the margin of sampling error to the nearest percent. $$ p=72 \%, n=100 $$
Short Answer
Expert verified
The margin of sampling error is approximately 9%.
Step by step solution
01
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.
02
Convert Percentage to Decimal
Convert the percentage \( p = 72\% \) to a decimal for calculation. This is done by dividing by 100. Thus, \( p = 0.72 \).
03
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 \( n = 100 \): \[ \sqrt{\frac{0.72 \times 0.28}{100}} = \sqrt{\frac{0.2016}{100}} = \sqrt{0.002016} \approx 0.0449 \]
04
Apply the Z-score
Multiply the standard error by the z-score for a 95% confidence level, which is 1.96. \[ ME = 1.96 \times 0.0449 \approx 0.088 \]
05
Convert Margin of Error to Percentage
Convert the margin of error back to a percentage by multiplying by 100. Therefore, \( ME = 0.088 \times 100 \approx 8.8\% \).
06
Round to Nearest Percent
Round the margin of error to the nearest whole percentage. Hence, the margin of error is approximately \( 9\% \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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
- 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.
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.
