Minting Dollar Coins For the sample data from Exercise 1, we get a P-value of 0.0041 when testing the claim that σ < 0.04000 g.


What should we conclude about the null hypothesis?
What should we conclude about the original claim?
What do these results suggest about the new minting process?

Verified step by step guidance

1. Step 1: Understand the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis (H₀) states that σ ≥ 0.04000 g, while the alternative hypothesis (H₁) states that σ < 0.04000 g. This is a one-tailed test for the population standard deviation.
2. Step 2: Interpret the P-value. The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true. Here, the P-value is 0.0041.
3. Step 3: Compare the P-value to the significance level (α). If the P-value is less than the chosen significance level (commonly α = 0.05), we reject the null hypothesis. In this case, 0.0041 < 0.05, so we reject H₀.
4. Step 4: Conclude about the original claim. Since the null hypothesis is rejected, there is sufficient evidence to support the claim that σ < 0.04000 g. This suggests that the variability in the minting process is less than 0.04000 g.
5. Step 5: Interpret the results in context. The results suggest that the new minting process is more consistent (less variable) than the threshold of 0.04000 g. This could indicate an improvement in the precision of the minting process.

Key Concepts

Null Hypothesis

The null hypothesis is a statement that there is no effect or no difference, and it serves as the default assumption in statistical testing. In this context, it posits that the standard deviation (σ) of the minting process is greater than or equal to 0.04000 g. A low P-value indicates strong evidence against the null hypothesis, suggesting it may be rejected in favor of the alternative hypothesis.

P-value

The P-value is a statistical measure that helps determine the significance of results in hypothesis testing. It represents the probability of observing the data, or something more extreme, assuming the null hypothesis is true. A P-value of 0.0041 indicates a very low probability of obtaining such results if the null hypothesis were correct, leading to the conclusion that the null hypothesis is likely not true.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. In this scenario, it would claim that the standard deviation (σ) is less than 0.04000 g. The results of the test, particularly the low P-value, provide evidence supporting the alternative hypothesis, indicating that the new minting process may be more precise than previously thought.