QBIO 305 Statistics for the Life Sciences

Let Y_i be independent and identically distributed random variables, each with mean \mu and standard deviation \sigma.
For large n:
- Mean: \bar{Y}_n = \frac{Y_1 + \cdots + Y_n}{n}
- Approximation: \bar{Y}_n \approx \mu + \frac{\sigma}{\sqrt{n}} N(0, 1)
- Where N(0, 1) is a standard normal random variable.
The Central Limit Theorem provides information about the average of these random variables.

Example: Weight of lambs is viewed as a random variable where:
- Population mean \mu and population standard deviation \sigma are unknown.
- Mean weight can be calculated from the sample, allowing estimation of the population parameters.

Case Study: Researchers captured 14 male Monarch butterflies and measured wing area:
- Sample Mean: \bar{y} = 32.8143 \approx 32.81 \, cm^2
- Sample Standard Deviation: s = 2.4757 \approx 2.48 \, cm^2
Population mean and standard deviation represented as:
- \mu = population mean wing area,
- \sigma = population SD of wing area.
Sample mean \bar{y} serves as an estimate for population mean \mu and sample SD s estimates \sigma.

Standard deviation of the sampling distribution of \bar{Y} is:
- \sigma_{\bar{Y}} = \frac{\sigma}{\sqrt{n}}
Using sample SD, standard error (SE) is:
- \text{SE}_{\bar{Y}} = \frac{s}{\sqrt{n}}
Example Calculation:
- For n = 28 lambs:
  - s = 0.65 \, kg
  - \bar{y} = 5.17 \, kg
- SE Calculation:
  - \text{SE} = \frac{0.65}{\sqrt{28}} \approx 0.12 \, kg

Standard Deviation (SD): Measures variability among individual observations in the sample.
Standard Error (SE): Measures variability associated with the sample mean as an estimate of the population mean.

Figures represent data displayed as \bar{y} \pm \text{SE}:
- Includes bar graphs and interval plots.
Importance of clear labeling as SE and SD figures differ significantly.

Given data set:
- Unknown values of population mean \mu and standard deviation \sigma.
- Sample mean \bar{y} and sample SD s calculated.
Approximation and distribution:
- From the Central Limit Theorem, \frac{\bar{Y} - \mu}{\sigma/\sqrt{n}} has a N(0, 1) distribution.
- Use of z-scores for 95% confidence interval:
  - P(-1.96 < \frac{\bar{Y} - \mu}{\sigma/\sqrt{n}} < 1.96) \approx 0.95
Calculate 95% CI:
- Use sample SD s in place of \sigma if normal distribution holds.

If data approximates normal distribution:
- Use Student's t distribution to construct confidence intervals.
With increasing degrees of freedom (df), the t-distribution approaches the normal distribution.

Two-sided 95% CI: Use 0.025 column.
One-sided confidence intervals: 90% lower (0.10) and upper (0.05) bounds calculated accordingly.

Given:
- n = 14, \bar{y} = 32.8143, s = 2.4757
95% Confidence Interval Calculation:
- \bar{y} \pm \left(t_{0.025} \times \frac{s}{\sqrt{n}}\right)
  - Result: 32.81 \pm \left(2.16 \times \frac{2.48}{\sqrt{14}}\right)
- Final interval: Approx (31.4, 34.2)
90% Confidence Interval Calculation:
- \bar{y} \pm \left(t_{0.05} \times \frac{s}{\sqrt{n}}\right)
- Result: Approx (31.6, 34)

Importance of checking normal distribution for valid confidence interval estimation.
Confidence interval does not imply population mean is probabilistically bounded but rather is an estimate.
If many independent samples taken and confidence intervals calculated, expect that about 95% will capture the true mean.
As sample size increases, interval width decreases (diminishing return phenomenon).

To estimate necessary sample size for desired SE:
- E.g., SE = rac{s}{\sqrt{n}} must meet a specific precision threshold.

Ensure the sample is a simple random sample to uphold the integrity of statistical inferences.
For small samples, data must be approximately normal. Larger samples (typically n=30) invoke Central Limit Theorem.
High awareness of possible outliers is required.