Hypothesis Testing in Statistics

Example Problem: Testing Cereal Box Weights

Step 1: Identify Hypotheses

  • Null Hypothesis (H0): The mean weight of cereal boxes is 20 ounces.

  • Alternative Hypothesis (H1): The mean weight of cereal boxes is less than 20 ounces (H1: mean < 20).

Step 2: Collect Data

  • Sample weights of cereal boxes (in ounces): 19.5, 19.8, 19.9, 19.7, 19.4

Step 3: Select Alpha Level

  • Set alpha (α) = 0.05.

Step 4: Calculate Test Statistics

  • Using a t-test, compute the sample mean and standard deviation:

    • Mean (X̄) = (19.5 + 19.8 + 19.9 + 19.7 + 19.4) / 5 = 19.66

    • Standard deviation (s) is calculated as 0.20.

    • Test statistic (t) = (X̄ - 20) / (s/√n) = (19.66 - 20) / (0.20/√5) = -3.77.

Step 5: Determine p-value

  • Using statistical software or a t-table, find the p-value corresponding to t = -3.77 and 4 degrees of freedom.

  • Assume the p-value is approximately 0.01.

Step 6: Make a Decision

  • Since p-value (0.01) < alpha (0.05), reject H0.

Step 7: Conclusion

  • There is sufficient evidence to conclude that the mean weight of the cereal boxes is less than 20 ounces.

The alpha level (α) is a predefined threshold set by researchers to determine the statistical significance of their test results. It represents the probability of making a Type I error, which occurs when the null hypothesis (H0) is incorrectly rejected. In simpler terms, it indicates how much risk a researcher is willing to take in claiming that a difference exists when it actually does not.

Common alpha levels include:

  • 0.05: This means there is a 5% chance of rejecting a true null hypothesis. It's commonly used in many scientific studies.

  • 0.01: This indicates a stricter threshold with only a 1% chance of making a Type I error.

Before executing the hypothesis test, researchers must select an alpha level to determine the significance of the results, aiding in the interpretation of the p-value obtained from the test. A p-value that falls below the alpha level reinforces the evidence against the null hypothesis, leading to its rejection, while a p-value above the alpha level suggests insufficient evidence to challenge the null hypothesis.