Problem 13 - Calculus for Biology and Medicine

Question

A certain study showed that less than \(5 \%\) of the population suffers from a certain disorder. To get a more accurate estimate of this proportion, you plan to conduct another study. What sample size should you choose if you want to be at least \(95 \%\) sure that your estimate is within \(0.05\) of the true value?

Step-by-Step Solution

Answer

A sample size of 74 is needed.

Step 1: Define the Parameters

First, we'll define the known parameters needed for the calculation. The critical z-value for a 95% confidence level is approximately 1.96. The proportion from the initial study is less than 5%, so we'll use an estimate of \( p = 0.05 \). The margin of error is specified as 0.05.

Step 2: Use the Sample Size Formula

The formula for determining sample size \( n \) when estimating a proportion is: \[ n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right) \]Where \( Z = 1.96 \) (z-value for 95% confidence), \( p = 0.05 \), and \( E = 0.05 \) (the margin of error).

Step 3: Plug Values into the Formula

Substitute the values into the formula:\[ n = \left( \frac{1.96^2 \cdot 0.05 \cdot (1 - 0.05)}{0.05^2} \right) \]This simplifies to: \[ n = \left( \frac{1.96^2 \cdot 0.05 \cdot 0.95}{0.0025} \right) \].

Step 4: Calculate the Sample Size

Calculate the values step-by-step. First compute: \( 1.96^2 = 3.8416 \). Then, calculate the expression:\[ n = \left( \frac{3.8416 \cdot 0.05 \cdot 0.95}{0.0025} \right) \] Finally, compute the full expression:\[ n \approx \left( \frac{0.18292}{0.0025} \right) \approx 73.168 \]Since sample size must be a whole number, round up to obtain \( n = 74 \).

Key Concepts

Confidence Interval

When you hear the term "confidence interval," think of it as a range of values that is used to estimate a certain population parameter, like a proportion. Instead of producing one single number, a confidence interval provides a span that likely contains the true value. This range is designed to cover the true parameter a certain percentage of the time, typically 95% or 99%.

• Confidence intervals are important because they provide more information than a single-point estimate. They offer insights into the reliability and precision of the estimate.
• For example, if you are 95% confident, it means you expect the interval to contain the true proportion 95 times out of 100.

This concept is especially useful when the exact population parameter is unknown, which is often the case in statistical analyses. In our exercise, the interval gives a range where we believe the true proportion of the disorder lies. Thus, a confidence interval helps us to express statistical uncertainty in a measured and quantifiable way.

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and a sample estimate. It is a crucial component for defining the width of a confidence interval.

• The margin of error is influenced by sample size and variability of the data. Larger samples generally lead to smaller margins of error, making estimates more precise.
• In our original exercise, the desired margin of error is 0.05. This means we want our estimate to be within 5% of the actual proportion of the population that suffers from the disorder.

By selecting a margin of error, we are essentially controlling how precise we want our estimate to be. A smaller margin of error demands a larger sample size but gives greater confidence in the closeness of our sample estimate to the true population value.

Proportional Estimation

Proportional estimation involves determining the proportion of a population that exhibits a particular characteristic or trait. In this context, the proportion is the fraction of the population suffering from a disorder.

• This form of estimation is key for understanding and quantifying populations when exact data is unavailable. It is commonly used in surveys and public health research.
• Since the initial study suggests that less than 5% of the population is affected, we begin by estimating this proportion at 5%.

For accurate proportional estimation, having a sample that correctly represents the population is crucial. Our goal in the exercise is to use the sample size formula to determine how many individuals we need to include in our study to ensure that our proportional estimate is within a specified margin of error.

Critical z-value

The critical z-value is a statistical measure that corresponds to the desired confidence level in a standard normal distribution. This value indicates how many standard deviations away from the mean your result is under the null hypothesis.

• For a 95% confidence level, the critical z-value is 1.96. This means approximately 95% of the data under a normal distribution falls within 1.96 standard deviations from the mean.
• The critical z-value plays a vital role in determining sample size and constructing confidence intervals because it scales the standard deviation to match the desired level of confidence.

In the context of our exercise, the z-value is used in the sample size determination formula to calculate how large our sample needs to be. This ensures that our estimate is reliably within the desired margin of error, guaranteeing that we're 95% confident in its accuracy.