Question

An experiment is being planned to compare the effects of several diets on the weight gain of beef cattle, measured over a 140 -day test period. ${}^{20}$ In order to have enough precision to compare the diets, it is desired that the standard error of the mean for each diet should not exceed $5\mathrm{kg}$. (a) If the population standard deviation of weight gain is guessed to be about $20\mathrm{kg}$ on any of the diets, how many cattle should be put on each diet in order to achieve a sufficiently small standard error? (b) If the guess of the standard deviation is doubled, to $40\mathrm{kg}$, does the required number of cattle double? Explain.

Short answer

For part (a), 16 cattle are needed for each diet to not exceed the desired SE of 5 kg. For part (b), the required number of cattle quadruples to 64, not doubles, when the guessed standard deviation is doubled to 40 kg.

Step-by-step solution

Understanding the Requirement for Standard Error

The standard error (SE) of the mean is given by the formula $SE=\frac{\sigma }{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. The problem states that the SE should not exceed 5 kg.

Solving for the Sample Size (n)

To find the sample size needed to achieve the desired SE, rearrange the formula to solve for $n$ and plug in the values of $\sigma =20$ kg and $SE=5$ kg: $n={\left(\frac{\sigma }{SE}\right)}^2={\left(\frac{20}{5}\right)}^2=16$.

Considering the Effect of Doubling the Standard Deviation

Doubling the standard deviation to $40$ kg would affect the sample size required to achieve the same SE. The new sample size is calculated using the modified standard deviation: $n={\left(\frac{\sigma }{SE}\right)}^2={\left(\frac{40}{5}\right)}^2=64$.

Comparing the Two Sample Sizes

Comparing the two sample sizes calculated for the two standard deviation guesses: With $\sigma =20$ kg, we calculated $n=16$, and with $\sigma =40$ kg, we calculated $n=64$. Even though the standard deviation was doubled, the required number of cattle did not double; it actually quadrupled.

Key concepts

Sample Size Calculation

Determining the right sample size is a cornerstone in quantitative research, particularly in experimental studies like the one described in our exercise. Calculating the sample size involves understanding the desired precision of the results. In this case, the desired accuracy is influenced by the standard error (SE), which should not exceed 5 kg for weight gain. To keep SE low, a larger sample size is typically required.

The sample size (n) is calculated using the formula: $$n={\left(\frac{\sigma }{SE}\right)}^2$$. As we've seen in the solution, with an estimated population standard of 20 kg, we need 16 cattle. Doubling the standard deviation requires adjusting the sample size to maintain the same standard error; thus, we quadruple the number of cattle to 64, not double, due to the squared relationship in the formula.

Population Standard Deviation

The population standard deviation ($\sigma$) is a measure of how much individual measurements in a population deviate from the population mean. It's vital for determining the variability of the data, which directly affects the required sample size for an experiment. A higher standard deviation indicates more variability, leading to a larger sample needed to achieve a certain level of precision in estimating the mean.
In our scenario, with $\sigma =20$ kg, less data is needed to reach the desired standard error compared to when $\sigma =40$ kg. It's also important to understand that the population standard deviation is an estimate, and a change in this estimate can have significant implications for the design of a study, as it dramatically alters the required sample size.

Quantitative Research Methodology

Quantitative research methodology is a structured way of collecting and analyzing data to quantify the problem and create statistical models to test hypotheses. It typically involves objective measurements and the statistical, mathematical, or computational analysis of data collected through polls, surveys, and questionnaires, or by manipulating pre-existing statistical data.

In our exercise, the quantitative research method would comprise several critical steps: forming a clear hypothesis about diets' effects on weight gain, planning the data collection procedure, such as deciding on the sample size and ensuring measurement precision, and finally, statistically analyzing the data to draw conclusions about the dietary effects. The standard error and population standard deviation play crucial roles in quantitative research in determining the reliability and generalizability of the experiment's outcomes.