Hypothesis Testing Example: One-Sample t-Test
Hypothesis testing is a statistical decision-making process used to evaluate claims about a population.
In this example, we conduct a one-sample t-test to verify a claim about the average soda volume in cans.
Scenario:
A soft drink company claims that the average amount of soda in its cans is 330 ml.
A customer suspects that the actual volume is less. A random sample of 30 cans is taken and measured.
Step 1: Define Hypotheses
Null Hypothesis (H0): The average soda volume is 330 ml (µ = 330 ml).
Alternative Hypothesis (Ha): The average soda volume is less than 330 ml (µ < 330 ml).
Step 2: Select the Significance Level (α)
We choose α = 0.05, meaning we allow a 5% probability of rejecting H0 when it is actually true.
Step 3: Sample Data
Sample Size (n) = 30
Sample Mean (X■) = 328 ml
Sample Standard Deviation (s) = 3 ml
Step 4: Compute the Test Statistic
Using the formula:
t = (X■ - µ) / (s / sqrt(n))
t = (328 - 330) / (3 / sqrt(30))
t = -3.66
Step 5: Critical Value and p-Value
Using a t-table, the critical value for df = 29 at α = 0.05 (one-tailed) is -1.699.
Our calculated t-value (-3.66) is lower, and the corresponding p-value is 0.0005.
Step 6: Make a Decision
Since t < critical value and p-value < α, we reject H0.
The evidence suggests the average soda volume is less than 330 ml.
Step 7: Conclusion
Based on this hypothesis test, we conclude that the soda cans contain less than the advertised 330 ml on average.
