To analyze this hypothesis test problem, let's go through the provided information and determine the steps required to calculate the p-value and make the correct decision.

Given Information

1. Hypotheses:

   • Null hypothesis ($H_0$): $\mu = 0.678$
   • Alternative hypothesis ($H_a$): $\mu \neq 0.678$ (This is a two-tailed test.)

2. Sample Size:

   • $n = 20$

3. Test Statistic (t):

   • $t_{TS} = 2.418$

4. Significance Level:

   • $\alpha = 0.01$

5. Z-Score:

   • $z = 2.46$ is also given.

6. Requirement:

   • We need to determine whether to use the t-distribution or the z-distribution.
   • Based on the sample size (20), typically, a t-distribution is used if the population standard deviation is unknown, and the sample size is relatively small (n < 30).

Steps to Solve

1. Determine the Appropriate Distribution:

   • Since the sample size is 20, we should use the t-distribution.
   • We need to find the p-value using the t-distribution with 19 degrees of freedom (df = $n - 1 = 20 - 1 = 19$).

2. Calculate the P-value:

   • Using the given $t_{TS} = 2.418$, we can look up or calculate the p-value for a two-tailed test with 19 degrees of freedom.
   • For a t-score of 2.418 with df = 19, the approximate p-value falls within the range given (0.05 < p < 0.10), meaning it does not meet the 0.01 significance level threshold.

3. Decision:

   • Since $\alpha = 0.01$, and our p-value is larger than 0.01, we do not reject $H_0$.

Final Answers

• P-value: Approximate range $0.05 < p < 0.10$
• Decision: Fail to Reject $H_0$ (despite the "Reject $H_0$" circle being marked, the correct decision would be to Fail to Reject).

Would you like more details on any part of this solution, or do you have further questions?

1. What is the process for calculating the p-value using the t-distribution?
2. How would the decision change if $\alpha$ was set to 0.05 instead of 0.01?
3. Why do we use a t-distribution instead of a z-distribution in this case?
4. How can we interpret the p-value range in relation to hypothesis testing?
5. What would happen if this were a one-tailed test?

Tip: Always consider the sample size and known parameters to choose the appropriate distribution (z or t) for hypothesis testing.