QBIO 305 Statistics for the Life Sciences

QBIO 305 Statistics for the Life Sciences Notes

Professor Zhengye Zhou

Central Limit Theorem

• 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.

Application of Central Limit Theorem

• 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.

Statistical Estimation

• 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 Error of the Mean

• 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 Error vs Standard Deviation

• 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.

Visual Representation of Data

• 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.

Confidence Interval for Population Mean $\mu$

• 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.

Student's t Distribution

• 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.

Critical Values of t Distribution

• Use t-tables for determining critical values based on degrees of freedom.

• Rounding down degrees of freedom may be necessary when values are not listed.

Confidence Interval Calculation for Different Levels

• Two-sided 95% CI: Use 0.025 column.

• One-sided confidence intervals: 90% lower (0.10) and upper (0.05) bounds calculated accordingly.

Butterfly Wing Area Example Calculation

• 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 $(31.6, 34)$

Normality Check and Interpretation

• 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).

One-Sided Confidence Intervals

• Lower bound calculated as: $\bar{y} - t_{0.10} \text{SE}_{\bar{Y}}$.

• Upper bound calculated as: $\bar{y} + t_{0.05} \text{SE}_{\bar{Y}}$.

Sample Size Estimation for Precision

• To estimate necessary sample size for desired SE:

  • E.g., $SE = rac{s}{\sqrt{n}}$ must meet a specific precision threshold.

Diminishing Returns in Data Collection

• Each additional observation improves accuracy less than previous ones.

Conditions for Validity of Estimation Methods

• 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.