The claim is that weights​ (grams) of quarters made after 1964 have a mean equal to 5.670 g as required by mint specifications. The sample size is nequals40 and the test statistic is tequalsnegative 3.123. Use technology to find the​ P-value. Based on the​ result, what is the final​ conclusion? Use a significance level of 0.05.

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.

---

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.

The claim is that weights​ (grams) of quarters made after 1964 have a mean equal to 5.670 g as required by mint specifications. The sample size is n equals 40 and the test statistic is t equals negative 3.123. Use technology to find the​ P-value. Based on the​ result, what is the final​ conclusion? Use a significance level of 0.05.

Learn how to calculate the P-value for a hypothesis test regarding the mean weight of quarters made after 1964. With a sample size of 40 and a test statistic of t = -3.123, find the conclusion at the 0.05 significance level. This problem involves hypothesis testing, t-distribution, and two-tailed tests.

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