WORK IN PROGRESS - Need to check the code Type I and Type II errors - (AI HL4.18) Normal distribution
Normal distribution Type I and Type II errors
WORK IN PROGRESS - Need to check the code
Scenario: The Case of Shrinking Potion Bottles
Background:
In the magical world of Alchemia, a potion master claims that each potion bottle contains precisely 1 liter of the Shrinking Potion. However, recent complaints from the wizarding community suggest that bottles might contain less than claimed.
Objective:
As an apprentice alchemist, you are tasked with investigating whether these claims are accurate using the principles of hypothesis testing and the normal distribution.
Investigation Steps:
1. Hypothesis Setting:
- Null Hypothesis (H0): The average volume of potion in a bottle is 1 liter (μ = 1).
- Alternative Hypothesis (H1): The average volume of potion in a bottle is less than 1 liter (μ < 1).
2. Data Collection:
- You decide to randomly sample potion bottles and measure their volumes.
- The sample mean will be compared to the claimed volume.
3. Statistical Analysis:
- Assume the sampling distribution of the mean follows a normal distribution.
- Calculate the sample mean and use the given standard deviation to compute the Z-score.
- Determine the critical value for a 5% significance level and compare it to the Z-score.
4. Decision Making:
- Based on the calculated Z-score and the critical region, decide whether to reject the null hypothesis.
- Discuss the consequences of making a Type I error (false alarm) or a Type II error (missed detection).
Questions for Investigation:
1. Discovery Question:
- What are some potential causes for the potion bottles not containing the full volume?
2. Understanding Significance:
- What does a 5% significance level mean in the context of this magical investigation?
3. Implications of Errors:
- If a Type I error occurs, what does that mean for the potion master's reputation?
- How would a Type II error affect the potion-buying wizards?
4. Reflection:
- Why is it important for potion masters to accurately measure the volume of their potions?
- How do the concepts of Type I and Type II errors apply in other areas of magical manufacturing?