Problem 20 - Algebra 2

Question

For Exercises \(12-21,\) find the margin of sampling error to the nearest percent. Three hundred sixty-seven of 425 high school students said pizza was their favorite food in the school cafeteria.

Step-by-Step Solution

Answer

The margin of sampling error is 3%.

Step 1: Identify the Proportion

First, identify the proportion of students who said pizza was their favorite food. The proportion, \( p \), can be found by dividing the number of students who preferred pizza by the total number of students surveyed. Here, \( p = \frac{367}{425} \).

Step 2: Calculation of Proportion

Calculate the value of \( p \) using the formula from the previous step. \( p = \frac{367}{425} \approx 0.8635 \). This means approximately 86.35% of the students prefer pizza.

Step 3: Calculate the Standard Error

The standard error (SE) of the proportion is calculated using the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}} \] where \( p = 0.8635 \) and \( n = 425 \).

Step 4: Substitute and Compute the Standard Error

Substitute \( p \) and \( n \) into the standard error formula: \[ SE = \sqrt{\frac{0.8635 \times (1 - 0.8635)}{425}} \approx \sqrt{\frac{0.1177}{425}} \] Compute \( SE \approx 0.0161 \).

Step 5: Determine the Margin of Error

The margin of error (ME) is typically found by multiplying the standard error by 1.96 (for a 95% confidence level): \[ ME = 1.96 \times SE \approx 1.96 \times 0.0161 \approx 0.0316 \].

Step 6: Convert Margin of Error to Percentage

Convert the margin of error into a percentage by multiplying by 100. \( ME \approx 0.0316 \times 100 \approx 3.16\% \).

Step 7: Round to the Nearest Percent

Finally, round the margin of error to the nearest percent. In this case, 3.16% rounds to 3%.

Key Concepts

Proportion

When dealing with statistical data, a proportion plays a crucial role as it represents the fraction of the total that meets a certain condition. In this case, we are interested in the proportion of high school students who reported pizza as their favorite food in the cafeteria. To find the proportion, simply divide the number of students who like pizza by the total number of students surveyed. Using our example, the proportion is calculated as \( p = \frac{367}{425} \). This gives us approximately 0.8635, or 86.35% of students preferring pizza. Understanding the proportion helps us to make sense of the sample data before making forecasts or conclusions.

Standard Error

The standard error of a proportion is essential for understanding the variability of a sample statistic. It gives us an idea of how much the proportion can vary from sample to sample. To calculate the standard error for a proportion, use the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}} \]where \( p \) is our sample proportion and \( n \) is the sample size. In our scenario, substituting the values gives: \[ SE = \sqrt{\frac{0.8635 \times (1 - 0.8635)}{425}} \]Computing this, we find \( SE \approx 0.0161 \). The smaller the standard error, the more reliable the sample mean is as an estimate of the population mean. A smaller SE suggests less variation and more precision in the estimate.

Confidence Level

A confidence level indicates how sure we are that a particular range (confidence interval) includes the true population parameter. It is typically expressed as a percentage, such as 95%. In our example, a 95% confidence level implies that, if we repeat our survey many times, then 95% of the estimated ranges will contain the actual population proportion. This is a standard level of confidence, offering a reasonable balance between precision and reliability. The z-score corresponding to a 95% confidence level is approximately 1.96, which is used to calculate the margin of error by multiplying it with the standard error.

Margin of Error

The margin of error quantifies the uncertainty in the estimation of the population proportion. It provides a range in which the true proportion is likely to fall. This is an important concept because it indicates the possible variation in our estimate if the survey were repeated.To calculate the margin of error, multiply the standard error by the z-score associated with your confidence level. For a 95% confidence level, as in our case, the z-score is 1.96, thus:\[ ME = 1.96 \times SE \approx 1.96 \times 0.0161 \approx 0.0316 \]The margin of error is then converted to a percentage (\( 0.0316 \times 100 = 3.16\% \)) and rounded to 3%. The margin of error helps in understanding the potential range of variation around the sample proportion, enhancing the interpretation of survey results.