Problem 19 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

In a survey, \(8 \%\) of adult Internet users said they participate in sports fantasy leagues online. The margin of error is \(\pm 4 \%\). Describe and correct the error in calculating the sample size. $$ \begin{aligned} \pm 0.08 &=\pm \frac{1}{\sqrt{n}} \\ 0.0064 &=\frac{1}{n} \\ n & \approx 156 \end{aligned} $$

Step-by-Step Solution

Answer

The mistake in the calculation is the misuse of the 8% proportion as the standard deviation in the margin of error calculation. When the correct formula is used for a 95% confidence level, the correct sample size comes out to be approximately 865, not 156.

Step 1: Identify the Mistake

The equation \(\pm 0.08 =\pm \frac{1}{\sqrt{n}}\) is incorrect. The given \(8\%\) is not the standard deviation but the proportion of users in the sample that participates in sports fantasy leagues online. The error seems to arise from wrongly taking \(8\%\) as the standard deviation, thus assuming it equal to \(\frac{1}{\sqrt{n}}\).

Step 2: Correct Formula for Margin of Error

The correct formula for margin of error in a proportion survey is \(E = z \sqrt{ \frac{p(1-p)}{n}}\), where \(E\) is the margin of error, \(z\) is the z-value from the normal distribution table, \(p\) is the estimated proportion (in this case 0.08 = 8%), and \(n\) is the sample size.

Step 3: Applying the Correct Formula

With a typical choice of \(95\%\) confidence, the z-value is \(1.96\). The equation to determine the sample size becomes: \(0.04 = 1.96 \sqrt{ \frac{0.08 \times 0.92}{n}}\). Solving this equation for \(n\) will yield the correct sample size.

Step 4: Calculate The Correct Sample Size

On solving the equation, the correct value of \(n\) is approximately 865, not 156.

Key Concepts

Survey Sample Size Calculation

Determining the appropriate sample size for a survey is critical to ensure that the results are statistically significant and reflect the views or behaviors of the population. The calculation of sample size depends upon several factors including the expected proportion, the desired confidence level, and the acceptable margin of error.

To start, let's understand the margin of error which is the range within which we expect our survey's results to reflect the true population value. A smaller margin of error requires a larger sample size, which delineates how confident we are about the results. The formula for calculating sample size (n) when estimated proportion (p) is involved is rooted in the equation: \[\begin{equation}E = z \sqrt{\frac{p(1-p)}{n}}\end{equation}\]where E represents the margin of error, z the z-value corresponding to the desired confidence level, and p the estimated proportion of the population with the characteristic of interest.

Using this formula, one can algebraically manipulate it to solve for n. This calculation assumes a simple random sample from a population and is part of a proportion survey, which studies the proportion of a population with a certain characteristic based on sample data.

Proportion Survey in Statistics

In statistical surveys, when we're interested in a certain characteristic within a population — such as the percentage of internet users participating in sports fantasy leagues — we implement what's known as a proportion survey. A critical aspect of this survey type is the proportion (p) which reflects the ratio of people in the sample with the characteristic to the total sample size.The challenge lies in ensuring that the sample proportion accurately signifies the population's proportion. Calculating the right sample size and employing a confidence interval are instrumental to that accuracy. The proportion p serves as a cornerstone in determining the margin of error, and subsequently, influences the sample size calculation.

A common mistake, as seen in our example, is confusing the proportion with other measures such as standard deviation, which can lead to significant errors in determining the sample size. Having clarity on these terms and their proper application is pivotal in statistics, especially when it comes to conducting reliable and valid surveys.

Z-Value and Confidence Interval

Now, let's delve into the concept of z-values and their role in confidence intervals. A z-value, often referred to as a z-score, is derived from the standard normal distribution. It represents the number of standard deviations a data point is from the mean.In the context of confidence intervals, the z-value is used to capture the range of values we are confident our population parameter lies within. Confidence intervals are all about balance—too narrow, and they may not include the true population parameter; too wide, and they become less useful. Choosing a z-value is a function of how confident we want to be that our interval covers the true parameter — it is intrinsically tied to our confidence level. Typically, a 95% confidence level—which indicates that if we were to take many samples, 95 out of 100 of those created intervals would contain the true population parameter—corresponds to a z-value of about 1.96.

Applying the correct z-value associated with the desired level of confidence is essential for accurate sample size calculation. As was improved in our exercise example, the use of the z-value corresponding to a 95% confidence level was integral to correcting the sample size estimate.