2026-02-03
A p-value measures the strength of evidence against the null hypothesis.
Definition:
\[p = P(\text{observing data this extreme} \mid H_0 \text{ is true})\]
Interpretation:
Key point: P-value is NOT the probability that \(H_0\) is true!
Research Question: Does vitamin C dose affect tooth growth in guinea pigs?
Experimental Design:
Variables:
len: Tooth length (numeric)dose: Vitamin C dose in mg/day (0.5, 1.0, 2.0)supp: Supplement type (OJ or VC)We use ANOVA (Analysis of Variance) to test if dose affects tooth length.
Hypotheses:
\[H_0: \mu_{0.5} = \mu_{1.0} = \mu_{2.0}\]
\[H_1: \text{At least one mean differs}\]
Where \(\mu_{0.5}\), \(\mu_{1.0}\), and \(\mu_{2.0}\) are the mean tooth lengths for each dose group.
Significance level: \(\alpha = 0.05\)
| Dose (mg/day) | Mean Length | Std Dev | Sample Size |
|---|---|---|---|
| 0.5 | 10.6 | 4.50 | 20 |
| 1 | 19.7 | 4.42 | 20 |
| 2 | 26.1 | 3.77 | 20 |
Observation: Mean tooth length increases dramatically with dose!
The boxplots clearly show increasing tooth length with higher doses.
Both supplement types show the same dose-response pattern.
Rotate the plot to explore the 3D relationship
anova_result <- aov(len ~ dose, data = ToothGrowth) summary(anova_result)
## Df Sum Sq Mean Sq F value Pr(>F) ## dose 2 2426 1213 67.42 9.53e-16 *** ## Residuals 57 1026 18 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Result: F-statistic = 67.42, p-value = 9.53e-16
With such a small p-value, we reject the null hypothesis
P-value = 0.000000000000000953 (essentially zero!)
Interpretation:
If dose truly had NO effect on tooth length, the probability of observing differences this large by random chance is less than 0.0000000000001%.
Conclusion: Vitamin C dose has a highly significant effect on tooth growth in guinea pigs. Higher doses produce substantially longer teeth.
This demonstrates the power of p-values in hypothesis testing