Sample Size Determination & Standard Error

Page 1 – Deriving the Required Sample Size ### Key Given Quantities - Population mean: $\mu$ - Population standard deviation: $\sigma$

(the numerical value is not stated explicitly, but the algebra will reveal it) - Desired standard error of the mean (SEM): $SE = 5$ - Unknown to determine: sample size $n$ ### Fundamental Formula The standard error of the mean is defined as

$SE = \frac{\sigma}{\sqrt{n}}$ ### Algebraic Solution for $n$ 1. Impose the target SEM:

$5 = \frac{\sigma}{\sqrt{n}}$ 2. Clear the denominator (multiply both sides by $\sqrt{n}$):

$5\sqrt{n} = \sigma$ 3. Isolate the square‐root term (divide both sides by $5$):

$\sqrt{n} = \frac{\sigma}{5}$ 4. Square both sides to solve for $n$:

$n = \left(\frac{\sigma}{5}\right)^2$ ### Numerical Result Because the final notes report $n = 36$, the implicit population standard deviation must have been

$\sigma = 30$

(since $5\times6 = 30$ and $\sqrt{36}=6$).

Therefore, a sample of 36 observations is required so that the sampling distribution of the mean will have a standard error of exactly $5$. ### Connections & Implications - The derivation hinges on the inverse‐square‐root relationship between SEM and sample size: doubling the sample size does not halve the SEM; SEM shrinks at the slower rate of $1/\sqrt{n}$. - In practical study design, knowing $\sigma$ (from pilot studies or historical data) lets researchers budget an adequate sample size before collecting data. - A larger $\sigma$ inflates the required $n$ quadratically; conversely, tolerating a larger SEM (looser precision) would reduce the necessary $n$ dramatically. ## Page 2 – Plain-Language Explanation & Intuition ### Everyday Analogy Imagine a city’s residents. You already know their overall average age $\mu$ and how much ages typically vary $\sigma$. To save effort, you will not interview everyone; you will select a smaller group (sample) and compute its average age. - Population – the entire city. - Sample – a manageable subset of residents. - Sample mean – the average age you find in that subset. - Standard error – how far that sample mean is likely to drift from the true city‐wide mean just by random chance. ### Why Controlling SEM Matters A smaller SEM means greater precision—your sample mean is likely to land close to $\mu$. Policymakers, quality‐control engineers, and scientists all care about this precision to make confident decisions. ### Relationship Re-stated The quantitative law

$SE = \frac{\sigma}{\sqrt{n}}$

shows: - Bigger $\sigma$ → bigger uncertainty (harder to estimate the mean precisely). - Bigger $n$ → smaller uncertainty (more information per average). ### Step-by-Step Arithmetic Recap 1. Start with $5 = \dfrac{\sigma}{\sqrt{n}}$. 2. Rearrange to $\sqrt{n} = \dfrac{\sigma}{5}$. 3. Square: $n = \left(\dfrac{\sigma}{5}\right)^2$. 4. Substitute $\sigma = 30$ → $n = \left(\dfrac{30}{5}\right)^2 = 6^2 = 36$. ### Practical Takeaway A sample of 36 people is large enough that, on average, the sample mean age will be within $\pm5$ years of the true population mean age. \n
• If you could only survey 9 people, the SEM would jump to $\dfrac{30}{\sqrt{9}} = 10$, doubling your expected error.

• Survey 144 people, and SEM would fall to $\dfrac{30}{12} = 2.5$, halving the error. ### Broader Relevance - Experimental Design – Clinical trials, manufacturing audits, and opinion polls all face the same balancing act: larger $n$ improves precision but costs more time and money. - Ethical Aspect – Oversampling wastes resources; undersampling risks misleading conclusions. Correct calculation aligns ethics with efficiency. - Statistical Foundation – This exercise exemplifies the Central Limit Theorem, which guarantees that