Statistical Concepts and Hypothesis Testing in Biology: Study Guide

Statistical Concepts in General Biology

Introduction

Statistical analysis is a fundamental tool in biology for interpreting data, testing hypotheses, and drawing conclusions about populations and experimental results. This study guide covers key statistical concepts relevant to general biology, including hypothesis testing, z-scores, and the central limit theorem.

Section 1: True or False Statements

Understanding Statistical Reasoning

• Standard Score (z-score): A z-score indicates how many standard deviations a data point is from the mean. Higher z-scores represent values further from the mean.

• Sample Size and Standard Error: Increasing the sample size decreases the standard error, making statistical tests more sensitive.

• Alpha Value (Significance Level): The alpha value (e.g., 0.01) sets the threshold for rejecting the null hypothesis. Lower alpha values require stronger evidence to reject the null hypothesis.

• One-tailed vs. Two-tailed Tests: Two-tailed tests require more extreme z-scores to reject the null hypothesis compared to one-tailed tests at the same alpha value.

Example: If you want to test whether a sample mean is significantly higher than a population mean, you might use a one-tailed test with an alpha of 0.05. If you want to test for any difference (higher or lower), you would use a two-tailed test.

Section 2: Central Limit Theorem

Key Principles

• Definition: The central limit theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's distribution.

• Factors: The theorem depends on sample size, population mean, and population standard deviation.

• Application: Allows biologists to use normal distribution-based statistics even when the underlying data is not normally distributed.

Example: If you measure the heights of 30 randomly selected plants, the average height will be approximately normally distributed even if individual heights are not.

Section 3: Calculating z-scores

Transforming Raw Scores

• z-score Formula: The z-score for an individual value is calculated as:

$z = \frac{X - \mu}{\sigma}$

• Application: Allows comparison of scores from different distributions or populations.

Name	Raw Score	z-score
Andy	4	Additional info: Calculate using population mean and standard deviation.
Brandi	10	Additional info: Calculate using population mean and standard deviation.
Claire	16	Additional info: Calculate using population mean and standard deviation.
Dieter	12	Additional info: Calculate using population mean and standard deviation.
Erica	8	Additional info: Calculate using population mean and standard deviation.

Example: If the mean score is 10 and the standard deviation is 4, Andy's z-score is $z = \frac{4-10}{4} = -1.5$.

Section 4: Hypothesis Testing

Steps in Hypothesis Testing

• State the null and alternative hypotheses using words or symbols.

• Determine whether the test is directional (one-tailed) or non-directional (two-tailed).

• Identify the critical value (z-critical) for the chosen alpha level.

• Calculate the test statistic (e.g., z-score).

• Decide whether to reject or fail to reject the null hypothesis.

• Explain the result in plain language.

Example: Testing whether a new keyboard increases typing speed: Null hypothesis (H0): No difference in typing speed. Alternative hypothesis (Ha): New keyboard increases typing speed.

Section 5: Types of Alternative Hypotheses

Directional vs. Non-directional Hypotheses

• Directional Hypothesis: Predicts the direction of the effect (e.g., higher, lower).

• Non-directional Hypothesis: Predicts a difference but not the direction.

Hypothesis	Type
COVID-19 vaccine group will have fewer symptoms	Directional (fewer)
Green juice will have a different weight	Non-directional (different)
Rustic Roast drinkers will be happier	Directional (happier)
Average apartment price is different	Non-directional (different)

Example: "People who drink green juice will have a different weight compared to the average" is a non-directional hypothesis.

Section 6: z-critical Values and Alpha Levels

Finding z-critical Values

• Alpha Level: The probability threshold for rejecting the null hypothesis (commonly 0.05 or 0.01).

• z-critical Value: The z-score corresponding to the chosen alpha level.

Alpha Level	Test Type	z-critical Value
0.05	Two-tailed	1.96
0.05	One-tailed (upper/right)	1.65
0.05	One-tailed (lower/left)	-1.65
0.01	Two-tailed	2.58
0.01	One-tailed	2.33

Example: For a two-tailed test with alpha = 0.05, you reject the null hypothesis if your z-score is greater than 1.96 or less than -1.96.

Section 7: Interpreting z-scores and Percentiles

Key Concepts

• Percentile: The percentage of scores below a given value.

• Finding Proportions: Use the z-score and standard normal table to find the proportion of the population below or above a score.

Example: If a z-score is 1.0, approximately 84% of the population scores below that value.

Section 8: Additional Concepts to Know

Summary of Key Skills

• How to interpret a z-score

• Relationship between mean, standard deviation, and z-score

• How to find proportions and percentiles using z-scores

• Difference between sample and population statistics

• Relationship between sample size and standard error

Additional info: These statistical concepts are foundational for analyzing biological data, designing experiments, and interpreting results in general biology.