Problem 21 - Algebra 2

Question

For Exercises \(12-21,\) find the margin of sampling error to the nearest percent. Nine hundred thirty-four of 2150 subscribers to a particular newspaper said their favorite sport was football.

Step-by-Step Solution

Answer

The margin of error is approximately 2%.

Step 1: Understand the Problem

We are asked to find the margin of error for a proportion. Given are 934 successes (subscribers who favor football) out of a total sample size of 2150.

Step 2: Calculate Sample Proportion

The sample proportion \( \hat{p} \) is calculated as the number of subscribers who favor football divided by the total number of subscribers, which is \( \hat{p} = \frac{934}{2150} \approx 0.434 \).

Step 3: Calculate Standard Error

The standard error for the sample proportion is given by \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \(n\) is the sample size. Substituting the values, we get \( SE = \sqrt{\frac{0.434(1-0.434)}{2150}} \approx 0.0107 \).

Step 4: Determine the Z-score for 95% Confidence

For a 95% confidence level, the Z-score is approximately 1.96. This is a commonly used value representing the Z-score for a 95% confidence interval.

Step 5: Calculate the Margin of Error

The margin of error (MOE) is calculated using the formula \( MOE = Z \times SE \). Thus, \( MOE = 1.96 \times 0.0107 \approx 0.021 \), or about 2.1% when rounded to the nearest percent.

Key Concepts

Sample Proportion

To understand sample proportion, think of it as a way to estimate how common a particular feature is within a larger population, based on a smaller group’s data, called a sample. In the context of our problem, we want to find out how many newspaper subscribers prefer football. Out of a sample size of 2150 subscribers, 934 expressed their love for football.

The formula for calculating the sample proportion (\( \hat{p} \)) is straightforward: divide the number of successes (people who prefer football) by the total number of individuals in the sample. For our data, the sample proportion comes out to be \( \hat{p} = \frac{934}{2150} \approx 0.434 \). This means approximately 43.4% of the sampled subscribers like football. This proportion helps in making general predictions about the entire subscriber base's preferences.

Standard Error

The standard error (SE) is like a measure of the 'wobbliness' of our sample estimate. It tells us how much the sample proportion might vary from sample to sample if we kept drawing different samples from the population. With our proportion of 43.4%, we use this to tell how accurately this figure might represent the population's preferences.

The formula to calculate standard error for a sample proportion is \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). Plugging our values into the formula, we find \( SE = \sqrt{\frac{0.434(1-0.434)}{2150}} \approx 0.0107 \).This means that the sample proportion of 43.4% could vary by about 1.07% due to sampling variability. This is a crucial aspect of understanding and interpreting survey results accurately.

Confidence Interval

Understanding confidence intervals is like understanding the range within which the true population parameter lies, with a certain probability. It gives you an interval to estimate if you repeated your survey several times. For instance, with a 95% confidence interval, you can be reasonably sure that if you took many samples, about 95% of the time, the interval will include the true proportion of the population.

To form a confidence interval, you'll need the sample proportion, standard error, and a Z-score, based on your desired confidence level. The margins of this interval are calculated as \( \hat{p} \pm Z \times SE \). Here, this means combining the sample proportion of 43.4% with a boundary formed by the product of the Z-score (1.96 for 95% certainty) and the standard error (0.0107). The result: a confidence interval of approximately 41.3% to 45.5%, giving us a good estimate of the preference for football in our subscriber population.

Z-score

The Z-score is a statistical measure that tells us how many standard deviations our sample proportion is from the mean of a population distribution. In survey estimates, it helps in determining how the values of the sample proportion relate to a normal distribution.

For a 95% confidence interval, the Z-score is commonly set at 1.96. This indicates that we are 95% confident that the interval formed contains the true proportion of the population who prefer football.

Why is this number critical? A higher confidence level means wider Z-score, causing a wider margin of error, while a lower confidence level does the opposite. So in practical terms, using a Z-score of 1.96 for our problem provides a balance between certainty and precision. It allows us to craft a reliable estimate while acknowledging the inevitable variability found in survey data.