The Chi Square Test

Chi Square Test

• The statistics section of the AP® Biology exam is considered one of the most difficult.

• Biology students often excel in memorization but find statistical analysis challenging.

• This article provides a systematic approach to mastering the Chi Square test.

1. Null and Alternative Hypothesis

Null Hypothesis (H₀):

• Predicts no effect of the independent variable on the dependent variable.

• Accepted if the data show no significant difference or pattern.

Alternative Hypothesis (Hₐ):

• Predicts an effect of the independent variable on the dependent variable.

• Accepted if data show significant differences or deviations from the null hypothesis.

Application to the coin flipping experiment

• Null Hypothesis (H₀):

  • The coin will land on heads 50% of the time and tails 50% of the time.

  • Predicts no change or deviation in the ratio of heads to tails due to external factors.

• Alternative Hypothesis (Hₐ):

  • The coin will not land on heads and tails evenly (not a 1:1 ratio).

  • Can hypothesize:

    • More heads than tails.

    • More tails than heads.

    • Any ratio different from 1:1.

Importance

• Creating null and alternative hypotheses ensures that results are framed within a testable and scientific context.

• Clear hypotheses allow scientists to interpret data and determine whether the null hypothesis should be accepted or rejected.

2. Defining the Variables

• χ2 (Chi Square):

  • Represents the value you calculate in the statistical test. Similar to solving for x in algebra, this is the main variable you aim to determine.

• O (Observed):

  • The quantitative data collected during the experiment. Examples include measurements like height, weight, or the number of occurrences (e.g., the number of heads or tails recorded in a coin-flipping experiment).

• E (Expected):

  • The values you predict before conducting the experiment, assuming the independent variable has no effect on the dependent variable. For a coin flip, the expected values would be an even split, such as 50 heads and 50 tails.

• ∑:

  • This is the sigma symbol or the summation. A symbol used to indicate "sum" in statistics. In this equation, it means adding together all the terms that follow it.

Application to the coin flipping experiment

• Let’s pretend that we performed the coin flip experiment and got the following data:

• Now we put these numbers into the equation:

  • Heads: (55-50)2/50 = .5

  • Tails: (45-50)2/50 = .5

  • Sum everything up: c2 =.5+.5 =1

Key Notes

• Equation Usage:

  • Essential for identifying whether observed data significantly differ from expected data.

  • Helps determine whether to accept or reject the null hypothesis.

• Exam Context:

  • The Chi Square equation will be provided on the AP® Biology Equations and Formulas sheet.

  • Memorization is not required, but understanding how to apply the equation is crucial.

• Common Challenge:

  • The equation may seem intimidating due to its format, but breaking it into smaller steps (define O, E, calculate differences, square them, divide by E, sum the values) simplifies the process.

3. Degrees of Freedom

• Degrees of Freedom (df):

  • Used to determine the threshold value needed for observed data to be significantly different from expected data.

  • Helps assess whether the null hypothesis should be accepted or rejected.

Application to the coin flipping experiment

• Number of Outcomes:

  • Two possible results: heads and tails.

• Degrees of Freedom Calculation:

  • df =2−1=1df = 2 - 1 = 1df =2−1=1.

• Using the Degrees of Freedom:

  • Use df to find the critical value from a chi-square table.

  • Compare the calculated chi-square value to the table value to determine statistical significance.

• Importance

  • Degrees of freedom ensure proper interpretation of statistical tests and guide scientists in determining the reliability of their results.

Outcome

• Calculated Chi Square Value: χ2=1\chi^2 = 1χ2=1.

• Critical Value (from table):

  • With df=1df = 1df=1, the critical value is 3.84.

Analysis

• Since χ2=1\chi^2 = 1χ2=1 is less than 3.84:

  • The observed data does not significantly differ from the expected data.

  • We fail to reject the null hypothesis.

Conclusion

• In this case, the data supports the null hypothesis:

  • The coin flip results align with the expectation of 50% heads and 50% tails.

Key Notes

• Calculation of Degrees of Freedom:

  • Formula: df=Number of outcomes−1df = \text{Number of outcomes} - 1df=Number of outcomes−1.

  • Indicates the flexibility or number of comparisons that can be made in the data.

AP Biology Exam Sample Question

• In a study examining fruit-fly behaviour, a covered choice chamber is used to determine whether the spatial distribution of flies is influenced by a substance placed at one end. To investigate their preference for glucose, 60 flies are introduced at the center of the chamber. A ripe banana is positioned at one end, while an unripe banana is placed at the opposite end. The number of flies in each section is observed and recorded after 1 minute and 10 minutes. Conduct a Chi-Square test using the data from the 10-minute time point, clearly stating the null hypothesis and concluding whether to accept or reject it.

• Begin by identifying the null hypothesis. The null hypothesis would be that the flies would be evenly distributed across the three chambers (ripe, middle, and unripe).

• Next, perform the Chi-Square test just like in the heads or tails experiment. Because there are three conditions, setting up the data might be useful.

• Chi Square: 48.9. 

• Our degrees of freedom are 3(ripe, middle, unripe)-1=2. 

• Let’s look at that table above for a confidence variable of .05 

• Value: 5.99. 

• Our Chi Square value of 48.9 is much larger than 5.99 so in this case we are able to reject the null hypothesis. This means that the flies are not randomly assorting themselves, and the banana is influencing their behaviour.

Summary

• This guide explores the Chi-Square test through two illustrative examples.

• Mastering the Chi-Square test requires practice, but once you understand the method, you can solve any Chi-Square problem using the same systematic approach.