Problem 19 - 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 survey of 32 people, 30 return a milk carton to the refrigerator immediately after using it.

Step-by-Step Solution

Answer

For the given question, (a) the sample proportion is 0.9375, (b) the margin of error needs to be calculated using the provided formula and assumed 95% confidence level, and (c) the interval likely to contain the true population proportion can be determined by subtracting and adding the calculated margin of error from/to the sample proportion.

Step 1 - Find the sample proportion

The sample proportion (p) is found by dividing the number of people displaying the characteristic (30 in this case) by the total sample size (32 in this case). So, \( p = 30 / 32 = 0.9375 \).

Step 2 - Find the margin of error

The margin of error (E) for a sample proportion is usually given by the formula: \( E = Z * \sqrt{ (p * (1 - p)) / n } \), where 'z' is the z-value from the standard normal distribution for the required confidence level (usually taken as 1.96 for a 95% confidence level), 'p' is the sample proportion and 'n' is the sample size. Since, no specific confidence level is mentioned, let's assume a 95% confidence level. So, \( E = 1.96 * \sqrt{ (0.9375 * (1 - 0.9375)) / 32 } \)

Step 3 - Find the interval

The confidence interval is then found by adding and subtracting the margin of error from the sample proportion. So, the interval is \( (p - E, p + E) = (0.9375 - E, 0.9375 + E) \).

Key Concepts

Margin of Error

The margin of error (MOE) is a crucial concept in statistics that helps define the range within which the true population proportion is likely to fall. It's essentially a measure of the precision of the estimate. The margin of error tells us how far off our sample results might be from the actual population value. To calculate it, we use the formula:

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

Here, \( Z \) is the z-value from the standard normal distribution, often 1.96 for a 95% confidence level. \( p \) is the sample proportion, and \( n \) is the sample size.
To better understand, imagine asking a group of people whether they put their milk back in the fridge. If 30 out of 32 say "yes", we calculate the sample proportion first and then apply the margin of error formula to see how much our findings might vary if we asked everyone. Understanding the margin of error helps us gauge how reliable our survey results might be.

Confidence Interval

The confidence interval is a statistical tool used to express the reliability of an estimate. It represents a range of values which we are fairly certain includes the true population proportion.
The formula for it is:

• Confidence Interval = \((p - E, p + E)\)

Here \( p \) is the sample proportion, and \( E \) is the margin of error. This interval provides an upper and lower bound around the sample proportion.
For example, with a sample proportion of 0.9375 and a calculated margin of error, the interval tells us where the true proportion is likely to lie. If constructed properly, it means we can be confident that in 95% of similar samples, the interval will capture the true population proportion. It gives both flexibility and assurance to those analyzing survey data, making it easier to communicate potential outcomes.

Population Proportion

The population proportion is a central figure in statistical surveys, reflecting how many subjects in a population exhibit a particular characteristic. Unlike the sample proportion, which is gathered from a subset, the population proportion represents the whole.
In our milk carton example, if we know the sample proportion is \( p = \frac{30}{32} = 0.9375 \), this acts as an estimate of the population proportion that we seek to uncover more broadly. However, due to practical constraints, actually measuring the true population proportion is often impossible, especially in large populations.
This is why sample surveys are conducted, allowing statisticians to make educated guesses about the population. Through tools like the margin of error and confidence intervals, we can provide well-informed estimates of the actual proportion across the entire group.