Problem 13 - Algebra 2

Question

For Exercises \(12-21,\) find the margin of sampling error to the nearest percent. $$ p=16 \%, n=400 $$

Step-by-Step Solution

Answer

The margin of sampling error is approximately 4%.

Step 1: Convert Percent to Decimal

First, convert the percentage value of the sample proportion \( p \) from a percentage to a decimal. Since \( p = 16\% \), divide by 100 to get \( p = 0.16 \).

Step 2: Calculate the Complement of Sample Proportion

Next, calculate the complement of the sample proportion, \( q = 1 - p = 1 - 0.16 = 0.84 \).

Step 3: Calculate the Standard Error

Use the formula for the standard error of a sample proportion: \( SE = \sqrt{\frac{p \cdot q}{n}} \). Substitute the values: \( SE = \sqrt{\frac{0.16 \cdot 0.84}{400}} \).

Step 4: Calculate the Margin of Error

The margin of error (ME) can be calculated as \( ME = Z \times SE \), where \( Z \) is the Z-score corresponding to the desired confidence level (usually 1.96 for 95%). For this exercise, since confidence interval is not specified, we'll assume 95%. Thus, \( ME = 1.96 \times SE \).

Step 5: Plug Values and Compute

Substitute the values and compute: \( SE = \sqrt{\frac{0.16 \cdot 0.84}{400}} = \sqrt{0.000336} \approx 0.0183 \). Then, \( ME = 1.96 \times 0.0183 \approx 0.035868 \).

Step 6: Convert Margin of Error to Percentage

Convert the margin of error from a decimal to a percentage by multiplying by 100: \( 0.035868 \times 100 \approx 3.59\% \). Round it to the nearest percent, which is \( 4\% \).

Key Concepts

Sample Proportion

The sample proportion is a foundational concept in statistics, especially when you're dealing with probability or populations. When statisticians conduct surveys or experiments, they often take a small portion of a population, called a sample, to infer characteristics about the whole group. In this process, the sample proportion comes into play. It is essentially the number of times an outcome occurs in a sample, divided by the total number of observations in that sample. For example:

• If you have 400 people surveyed and 16% like chocolate, your sample proportion would be 0.16 after converting the percentage to a decimal.

Converting the percentage to a decimal means dividing by 100. In this case, 16% becomes 0.16. The sample proportion is crucial because it serves as the basis for calculating other statistical measures like the standard error and confidence intervals.

Standard Error

The standard error provides insight into how much variability there is in a sample statistic. In the context of the sample proportion, the standard error measures the expected variation of the sample proportion from the true population proportion. The formula often used to calculate the standard error of a sample proportion is:\[ SE = \sqrt{\frac{p \cdot q}{n}} \]where:

• \( p \): sample proportion
• \( q \): the complement of the sample proportion \( (1-p) \)
• \( n \): total number of observations in the sample.

This formula derives from the principle that the more observations in your sample, the less variability you expect in the sample statistic. For instance, if you're looking at a proportion of 0.16 with 400 observations, your standard error will help determine how closely your sample proportion might reflect the true population proportion.

Confidence Interval

A confidence interval provides a range that is likely to contain the true value of a population parameter, such as the population proportion. The range is calculated so that it has a certain level of confidence, usually expressed as a percentage.Here's a step-by-step glimpse into why it's essential:- You start with a sample proportion and standard error.- Apply the standard error to calculate the range that estimates the population proportion.- This calculation often involves a confidence level, such as 95%, which means if you were to draw many samples, the interval would contain the true population parameter 95 times out of 100.The confidence interval is calculated as:\[ CI = p \pm (Z \times SE) \]where \( Z \) is the Z-score corresponding to the desired confidence level.Thus, a 95% confidence interval helps provide a buffer around your sample estimate, indicating where the true population proportion likely falls.

Z-score

The Z-score is a measure that indicates how many standard deviations an element is from the mean of a distribution. In the context of confidence intervals, a Z-score represents the number of standard errors you can expect a sample statistic to be from the true population parameter, given a specific confidence level. For common confidence intervals:

• A 95% confidence level uses a Z-score of 1.96.
• A 99% confidence level uses a Z-score of 2.576.

Why is this important? - The Z-score helps determine the width of the confidence interval. The higher the confidence level, the larger the Z-score, and thus, the wider the interval. - A larger Z-score or confidence level suggests more certainty that the interval contains the true parameter but results in a broader interval. Thus, using the appropriate Z-score for your confidence interval is essential to balance precision and confidence in your statistical analysis.