To solve this problem, let's break it down into steps:

Step 1: Set up the hypothesis

1. Null hypothesis (H₀): The mean weight of quarters made after 1964 is 5.670 grams. $H_0: \mu = 5.670$
2. Alternative hypothesis (H₁): The mean weight of quarters made after 1964 is not 5.670 grams (two-tailed test). $H_1: \mu \neq 5.670$

Step 2: Identify the given information

• Sample size, $n = 40$
• Test statistic, $t = -3.123$
• Significance level, $\alpha = 0.05$

Step 3: Find the P-value

Since this is a two-tailed test, the P-value is the probability of observing a test statistic as extreme as $t = -3.123$ in either direction (both tails of the distribution).

Using a t-distribution with $n - 1 = 40 - 1 = 39$ degrees of freedom, let's calculate the P-value for $t = -3.123$.

Using technology (like a t-distribution calculator or statistical software):

1. Look up the cumulative probability for $t = -3.123$ with 39 degrees of freedom.
2. For a two-tailed test, multiply this probability by 2.

After calculating, we find that:

$\text{P-value} \approx 0.0035$

Step 4: Make the conclusion

With a P-value of approximately 0.0035, we compare it to the significance level $\alpha = 0.05$:

• Since $\text{P-value} < 0.05$, we reject the null hypothesis.

Final Conclusion

There is enough evidence to conclude that the mean weight of quarters made after 1964 is different from 5.670 grams at the 0.05 significance level.

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Would you like further details on any part of this process? Here are some related questions to deepen your understanding:

1. How do we interpret a two-tailed test compared to a one-tailed test?
2. Why is the sample size relevant in hypothesis testing?
3. What is the role of the degrees of freedom in calculating the P-value?
4. How would our conclusion change if the significance level was 0.01?
5. What steps are necessary to perform this test manually without technology?

Tip: In hypothesis testing, always check if your test is one-tailed or two-tailed, as this affects the P-value calculation.