Question

Suppose you are planning an experiment to test the effects of various diets on the weight gain of young turkeys. The observed variable will be \(Y=\) weight gain in 3 weeks (measured over a period starting 1 week after hatching and ending 3 weeks later). Previous experiments suggest that the standard deviation of \(Y\) under a standard diet is approximately \(80 \mathrm{~g} .^{23}\) Using this as a guess of \(\sigma\), determine how many turkeys you should have in a treatment group, if you want the standard error of the group mean to be no more than (a) \(20 \mathrm{~g}\) (b) \(15 \mathrm{~g}\)

Short answer

To ensure the standard error does not exceed (a) 20 grams, at least 16 turkeys are needed; and for (b) 15 grams, at least 29 turkeys are needed.

Step-by-step solution

Understand the Relationship Between Standard Error, Standard Deviation, and Sample Size

The standard error of the mean (SEM) is related to the standard deviation of the population \(\sigma\) and the sample size \(n\) by the formula SEM = \(\frac{\sigma}{\sqrt{n}}\). The goal is to find the sample size \(n\) such that the SEM is not more than a specific value.

Solve for the Sample Size for Part (a)

Using the given condition that the standard error should be no more than 20 grams, set up the inequality \(\frac{80}{\sqrt{n}} \leq 20\) and solve for \(n\).

Calculate the Required Sample Size for Part (a)

Square both sides of the inequality \(\frac{80}{\sqrt{n}} \leq 20\) to get \(\frac{6400}{n} \leq 400\). Multiply both sides by \(n\) and divide by 400 to find \(n \geq \frac{6400}{400}\).

Round up to the Next Whole Number for Part (a)

Since the number of turkeys has to be a whole number, round up the result from Step 3 to the nearest whole number to ensure the standard error does not exceed 20 grams.

Solve for the Sample Size for Part (b)

Using the condition that the standard error should be no more than 15 grams, set up the inequality \(\frac{80}{\sqrt{n}} \leq 15\) and solve for \(n\).

Calculate the Required Sample Size for Part (b)

Square both sides of the inequality \(\frac{80}{\sqrt{n}} \leq 15\) to get \(\frac{6400}{n} \leq 225\). Multiply both sides by \(n\) and divide by 225 to find \(n \geq \frac{6400}{225}\).

Round up to the Next Whole Number for Part (b)

Similar to Step 4, round up the result from Step 6 to the nearest whole number to ensure the standard error does not exceed 15 grams.

Key concepts

Sample Size Determination

When conducting any experiment, it is critical to establish the appropriate number of observations or subjects necessary to achieve reliable results without unnecessary resource expenditure. This process is known as sample size determination. To illustrate, let's revisit the context of the example problem where an experiment is designed to measure the weight gain of young turkeys under various diets.

The predetermined criteria for the standard error of the group mean informs the minimum sample size required. This ensures that the average weight gain recorded is a precise estimate of the true population mean with an acceptable level of precision. In our turkey experiment, we seek sample sizes that yield a standard error of either 20g or 15g, translating into different sample size requirements for each scenario.

Improving the Experiment's Efficiency

Larger sample sizes usually provide more reliable data, but they also demand more resources. Therefore, it is a crucial balancing act between precision and practicality. Opting for the largest sample size might not always be feasible, so determining the smallest sample size that provides an acceptable standard error is beneficial. This not only conserves resources but also allows for a more efficient experimental design, avoiding the undue burden of assessing more subjects than necessary.

Furthermore, it's important to account for potential dropouts or unusable data, by slightly increasing the sample size beyond the minimum calculated. This precaution helps ensure adequate data is available when the experiment concludes.

Standard Deviation

The standard deviation is a statistical measure representing the amount of variability or dispersion in a set of values. In the context of the turkey experiment, it indicates how much individual turkey weight gains deviate from the average weight gain.

The value of the standard deviation, denoted as gives us insight into the spread of our data points from the mean. If the standard deviation is small, it suggests that the data points are closely clustered around the mean. Conversely, a high standard deviation indicates that the data are spread out over a wider range of values, implying more variability in turkeys' responses to their diets.

Role of Standard Deviation in Sample Size Calculation

Understanding standard deviation is crucial when determining sample size because it directly influences the standard error. As seen in the solution steps, the proposed standard deviation of 80g under a standard diet becomes a pivotal point in calculating the minimum number of turkeys needed. By knowing the variability in the population, researchers can ensure that their sample is representative and their estimates of the mean are accurate within the desired error margin.

Experimental Design in Statistics

Lastly, the concept of experimental design in statistics refers to the framework that guides the setup of an experiment to ensure that the right type of data is collected, the method of analysis is appropriate, and the results are valid. It encompasses everything from the selection of the sample and random assignment of treatments to the statistical tests used for data analysis.

In the turkey weight gain example, a proper experimental design would involve randomly assigning individual turkeys to different treatment groups, such as various diets, to eliminate biases and establish causation. Researchers must also consider other design aspects like blinding, which helps prevent subjective bias, and replication, which is the repetition of an experiment to validate findings.

Consequences of Poor Experimental Design

If the experimental design is flawed, it may lead to inaccurate or uninterpretable results. This could occur if, for instance, the sample size does not adequately represent the population or if confounding factors are not controlled. Therefore, careful planning in the experimental design stage is fundamental to ensure that the conclusions of an experiment are credible and valuable.

Effective experimental design considers the objectives of the research, the resources available, and employs appropriate statistical methods to answer the research question. The precision in sample size determination and understanding of standard deviation are just a few components of a broader experimental design process that ultimately serves to enhance the validity and reliability of an experiment's results.