Problem 18 - Algebra 2

Question

Surveys For each sample, find (a) the sample proportion, (b) the margin of error, and (c) the interval likely to contain the true population proportion. Of 500 teenagers surveyed, 460 would like to see adults in their community do more to solve drug problems.

Step-by-Step Solution

Answer

The solutions to the asked parts of the problem are: (a) the sample proportion (p) is \( \frac{460}{500} \) . (b) The margin of error (E) has been calculated using Z-value for 95% confidence level. (c) The confidence interval is calculated by (p - E, p + E). It is the range likely to contain the true population proportion.

Step 1: Calculate Sample Proportion

The sample proportion (p) is the number of successes divided by the number of trials. In this case, the number of 'successes' is the number of teenagers who would like to see adults do more to solve drug problems, which is 460. The number of trials is the total number of teenagers surveyed, which is 500. Therefore, the sample proportion (p) is calculated as \( p = \frac{460}{500} \) .

Step 2: Calculate Margin of Error

The formula to calculate the margin of error (E) for a proportion is given by \( E = Z \sqrt{ \frac{p(1-p)}{n} } \) where Z is the Z-value (which is 1.96 for a 95% confidence level), p is the sample proportion, and n is the sample size. Since we are considering a 95% confidence level, \( E = 1.96 \sqrt{ \frac{p(1-p)}{n} } \) .

Step 3: Determine Confidence Interval

The confidence interval is a range likely to contain the true population proportion. It is calculated by (p - E , p + E ) . After plugging the calculated values, we can get the final confidence interval.

Key Concepts

Understanding Sample Proportion

The sample proportion, often symbolized as \( p \), represents the fraction of the sample exhibiting a particular desired attribute. In the context of our exercise, out of 500 surveyed teenagers, 460 expressed that they want adults to address drug problems more actively. The sample proportion depicts this information by dividing the number of teenagers who responded positively by the total number surveyed. Thus, the formula is given as \( p = \frac{460}{500} \), which simplifies to \( p = 0.92 \). This means 92% of the surveyed teenagers hold the same viewpoint.
The sample proportion is a critical parameter in statistics because it is often used to estimate what fraction of a larger population would likely present the same attribute based on the survey's findings.
It's important to remember that the sample proportion is not the actual population proportion, but an attempt to approximate it using the gathered data from the surveyed group.

Decoding Margin of Error

The margin of error outlines how much we should expect the figures from a sample to differ from the actual population parameters. To compute the margin of error (\( E \)), we use the formula \( E = Z \sqrt{ \frac{p(1-p)}{n} } \).
Here's what each symbol represents:

• \( Z \) is the Z-value correlated with the desired confidence level. For a 95% confidence level, \( Z \) is 1.96. This value evolves as the confidence desired changes.
• \( p \) is the sample proportion previously calculated.
• \( n \) denotes the sample size, which is 500 for the given example.

By plugging in the values, we derive a numerical result signifying the expected range within which the sample proportion could vary due to random sampling errors.
Margin of error helps communicate the certainty of the data collected in surveys. Lower margin of errors typically imply better precision and closeness to the true population values.

Explanation of Confidence Interval

A confidence interval is the statistical construct computed to estimate the range in which the true population parameter may reside. For our scenario, the confidence interval offers a range likely to contain the true proportion of all teenagers who would like community adults to handle drug issues.
The confidence interval is defined by the formula \( (p - E , p + E) \), where \( p \) is the sample proportion and \( E \) is the margin of error.
Let's assume we've calculated our margin of error based on the prior section. Plugging these into the equation will give us two endpoints forming the interval. This interval signifies that we are 95% confident that the true population proportion lies within this range.
The wider the confidence interval, the less certain we are about being close to the actual population parameter, which encourages the importance of sample size and reducing variability for better survey accuracy.