Problem 26 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Consider a large population in which \(\rho\) percent (in decimal form) have a certain characteristic. To be reasonably sure that you are choosing a sample that is representative of a population, you should choose a random sample of \(n\) people where $$ n>9\left(\frac{1-\rho}{\rho}\right) \text {. } $$ a. Suppose \(\rho=0.5\). How large does \(n\) need to be? b. Suppose \(\rho=0.01\). How large does \(n\) need to be? c. What can you conclude from parts (a) and (b)?

Step-by-Step Solution

Answer

For \(\rho = 0.5\), we require a minimum value of \(n > 9(2) = 18\). For \(\rho = 0.01\), we need a minimum value of \(n > 9(99) = 891\). Hence, as \(\rho\) decreases, the required size of \(n\) for the sample to be representative of the population significantly increases.

Step 1: Calculate n for \(\rho=0.5\)

To find the sample size for when \(\rho=0.5\), plug \(\rho=0.5\) into the formula \(n > 9\left(\frac{1-\rho}{\rho}\right)\), and solve for \(n\).

Step 2: Calculate n for \(\rho=0.01\)

To find the sample size for when \(\rho=0.01\), plug \(\rho=0.01\) into the formula \(n > 9\left(\frac{1-\rho}{\rho}\right)\), and solve for \(n\).

Step 3: Reflect on the results

Look at the results from Step 1 and Step 2 and compare how the value of \(n\) changes as the value of \(\rho\) changes. Make observations about the trend.

Key Concepts

Statistical Sampling

Statistical sampling is a crucial process employed in statistics to make inferences about a large population based on a smaller, manageable number of observations known as a sample. The essence of sampling lies in its ability to save time and resources while still providing valuable information about a population.

When conducting a statistical survey, it's often impractical or impossible to study an entire population, especially if it is large or inaccessible. Therefore, a subset or sample is selected to represent the population. The key is to ensure that the sample adequately reflects the characteristics of the overall group, which is why various sampling methods such as simple random sampling, stratified sampling, or cluster sampling are used. These methods aim to minimize biases and provide a clear picture of the population's attributes without examining every individual.

A well-designed sampling process must consider potential sources of error and take steps to reduce them, ensuring that the results are as accurate as possible. The reliability of the findings from a sample depends greatly on the sampling technique and the sample size.

Representative Sample

A representative sample accurately mirrors the diverse characteristics of a population, allowing for conclusions to be drawn about the population from that sample. Securing a representative sample is essential to ensure demographic, geographical, behavioral, and other pertinent attributes of the total population are captured proportionately in the sample.

To achieve a representative sample, random selection methods are crucial as they help to avoid selection bias, where certain groups might be overrepresented or underrepresented. Every individual in the total population must have an equal chance of being included. For instance, if you were analyzing consumer behavior in a city, your sample should reflect the varied age, income levels, genders, ethnic backgrounds, and other significant factors present in the city's population.

Exercise Improvement Advice

When improving exercises on representative sampling, it’s important to involve diverse scenarios displaying various population traits. Including examples with different population sizes and demographics provides a wealth of context, helping students to grasp the importance and impact of representative sampling.

Sample Size Calculation

Sample size calculation is a statistical tool used to determine the number of observations or replicates needed in a sample to confidently infer about a population parameter. An appropriately sized sample is large enough to be statistically valid but small enough to be practical and cost-effective.

The formula provided in the exercise, \[ n > 9\left(\frac{1-\rho}{\rho}\right) \] is a simplified rule that aims to ascertain a reasonably adequate sample size, dependent on the proportion of the population having a certain characteristic (denoted by \(\rho\)). In part a, plugging in \(\rho=0.5\), we find that the sample size can be relatively small, while for part b, where \(\rho=0.01\), a much larger sample is required for the same level of confidence. This indicates that when a characteristic is rare within a population, a greater number of people need to be sampled to reliably capture that characteristic's presence.

Understanding how to calculate the proper sample size helps ensure that the collected data is robust and that the resulting statistics have the desired level of precision. It also prevents wasting resources on excessively large samples or making inaccurate conclusions from samples that are too small.