Problem 19 - Algebra 2

Question

For Exercises \(12-21,\) find the margin of sampling error to the nearest percent. In a recent survey, 431 full-time employees were asked if the Internet has made them more or less productive at work. 27\(\%\) said it made them more productive.

Step-by-Step Solution

Answer

The margin of sampling error is approximately 4%.

Step 1: Understanding the Problem

We are tasked with finding the margin of sampling error for a survey of 431 full-time employees, where 27% reported the Internet made them more productive.

Step 2: Identifying the Formula

The margin of sampling error can be found using the formula for margin of error in a proportion: \( E = Z \times \sqrt{\frac{p(1-p)}{n}} \), where \( E \) is the margin of sampling error, \( p \) is the sample proportion, \( n \) is the sample size, and \( Z \) is the z-score for the desired confidence level.

Step 3: Substitute Known Values

From the problem, we know \( p = 0.27 \) and \( n = 431 \). We will assume a 95% confidence level, for which \( Z \) is approximately 1.96.

Step 4: Calculate the Standard Error

The standard error (SE) is calculated as \( SE = \sqrt{\frac{0.27(1-0.27)}{431}} \). Calculate this by evaluating inside the square root first.

Step 5: Evaluate the Margin of Error

Substitute the standard error into the margin of error formula: \( E = 1.96 \times SE \). Calculate the margin of error, and then round to the nearest percent.

Step 6: Conclude with the Result

The margin of error rounded to the nearest percent gives us the final answer.

Key Concepts

Survey Data Analysis

Survey data analysis involves the interpretation and comprehension of data collected from surveys. It is a crucial step in understanding what the gathered information signifies and how it can be applied. Surveys, like the one mentioned with 431 full-time employees, often seek to gather opinions or facts to assist in decision-making or drawing conclusions.

• Identify the survey objectives clearly to ensure precise data collection.
• Analyze the collected data to draw relevant insights, such as the proportion of employees feeling more productive due to the Internet.
• Utilize statistical techniques to summarize and make sense of the data, ensuring that the results are both reliable and valid.

Being thorough in survey data analysis ensures that the conclusions drawn are representative and actionable.

Confidence Level

The confidence level in statistics indicates the percentage of all possible samples that can be expected to include the true population parameter. It's a way to measure how confident we are that the survey results reflect the real world. In our example, a 95% confidence level is assumed.

• This suggests that if the survey were repeated multiple times, 95% of the results would contain the actual proportion of employees who feel more productive.
• A higher confidence level means more certainty in the results but generally results in a wider margin of error.
• Common confidence levels are 90%, 95%, and 99%, with 95% being frequently used in practice.

Choosing the right confidence level is essential, balancing between certainty and precision.

Statistical Calculations

Statistical calculations are essential for deriving meaningful insights from survey data. They ensure that the methodology is sound and the interpretations are valid. For this exercise:

• The margin of error was calculated using the formula: \( E = Z \times \sqrt{\frac{p(1-p)}{n}} \).
• The standard error was also a crucial step: \( SE = \sqrt{\frac{0.27(1-0.27)}{431}} \).
• These calculations help determine how much the survey results might differ from the actual population values.

This ensures that when we report statistics like 27% of employees feeling more productive, we understand the possible variation in these numbers depending on the sample's luck.

Sample Size and Proportion

Sample size and proportion are fundamental concepts in statistics that influence the reliability and validity of survey results.

• Sample size refers to the number of respondents or observations included in a survey, in this case, 431 employees.
• A larger sample size generally reduces the margin of error, leading to more precise results.
• The sample proportion denotes the fraction or percent of the sample responding in a particular way, like 27% considering themselves more productive.

In your analysis, making sure the sample size is sufficiently large and random helps ensure that the sample proportion accurately reflects the wider population.