Problem 17 - 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. In a random sample of 408 grocery shoppers, 258 prefer one large trip per week to several smaller ones.

Step-by-Step Solution

Answer

The sample proportion is approximately 0.632. The margin of error is around 0.048. Therefore, there is a 95% chance that the interval 0.584 to 0.680 contains the true population proportion.

Step 1: Finding the Sample Proportion

The sample proportion (p) is calculated as the ratio of the number of people in the sample who prefer one large trip per week to the total number of people in the sample. The formula for calculating it is: \(p = x / n\), where x is the number of successes and n is the total number of observations. Therefore, \(p = 258 / 408\).

Step 2: Calculating the Margin of Error

The margin of error is a range of values above and below the sample statistic in a confidence interval. It's usually defined for a confidence level of 95%. The formula for calculating it is: \(E = Z * sqrt((p(1 - p)) / n)\), where p is the sample proportion, n is the total number of observations and Z is Z value for the desired confidence level. Z is 1.96 for 95% confidence. So, \(E = 1.96 * sqrt((0.6323 * (1 - 0.6323)) / 408)\).

Step 3: Finding the Confidence Interval

The confidence interval likely to contain the true population proportion is given by the formula: \(CI = p ± E\), where p is the calculated sample proportion and E is the margin of error. Therefore, the range of values is given by: \(CI = 0.6323 ± E \).

Key Concepts

Sample Proportion

When analyzing data from surveys or experiments, knowing the sample proportion is crucial. The sample proportion provides a good estimate of what is happening within your data. It's calculated by taking the ratio of the observed "successes" to the total number of observations within the sample.
For example, in a survey of 408 grocery shoppers, if 258 prefer one large shopping trip per week, the sample proportion (\(p\)) can be computed using the formula:

• \(p = \frac{x}{n}\)

Here, \(x\) represents the number of successes, and \(n\) is the total number of observations. Plugging in the values, we get \(p = \frac{258}{408}\).
This results in a sample proportion that reflects how a part of the sample prefers larger shopping trips. By understanding the sample proportion, researchers can make inferences about the sample and understand distribution trends.

Margin of Error

Understanding the margin of error is key in interpreting survey results. The margin of error gives a range, showing how much the sample results might differ from the actual population values.
For a 95% confidence level, use the formula to calculate the margin of error:

• \(E = Z \times \sqrt{\frac{p(1-p)}{n}}\)

Here, \(E\) is the margin of error, \(Z\) is the Z-value (1.96 for 95% confidence), \(p\) is the sample proportion, and \(n\) is the sample size. This provides an interval indicating the reliability of survey results.
For example, if the sample proportion is 0.6323 and the total sample size is 408, you compute the margin of error as \(E = 1.96 \times \sqrt{\frac{0.6323 \times (1-0.6323)}{408}}\). This calculation helps quantify uncertainty in your estimates.

Confidence Interval

A confidence interval provides a range of values, within which the true population proportion is likely to fall. This gives a clearer picture of variability and certainty in sample data.
The confidence interval is determined by adding and subtracting the margin of error from the sample proportion:

• \(CI = p \pm E\)

Where \(p\) is the sample proportion and \(E\) is the margin of error. This results in a lower and upper bound, indicating where the true population proportion likely resides.
For instance, if our sample proportion is 0.6323 and margin of error was calculated previously, then the confidence interval can be expressed as: \(CI = 0.6323 \pm E\). Utilizing the confidence interval aids in making well-informed assertions about the population from which the sample was drawn.