To determine whether the mean tuition and fees for private colleges in California differs from $35,000, we perform a hypothesis test using the given data and compute the P-value. Here's how it’s done:

---

Step 1: Hypotheses

• Null Hypothesis ($H_0$): $\mu = 35,000$ (the mean tuition is $35,000).
• Alternative Hypothesis ($H_a$): $\mu \neq 35,000$ (the mean tuition is not $35,000).

This is a two-tailed test.

---

Step 2: Test Statistic

We use the formula for the t-test statistic for a single sample: $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$ where:

• $\bar{x} = 38,000$ (sample mean),
• $\mu = 35,000$ (hypothesized mean),
• $s = 7,900$ (sample standard deviation),
• $n = 13$ (sample size).

Substituting the values: $t = \frac{38,000 - 35,000}{7,900 / \sqrt{13}} = \frac{3,000}{7,900 / 3.605} = \frac{3,000}{2,191.6} \approx 1.37$

---

Step 3: Degrees of Freedom

Degrees of freedom ($df$) for a single-sample t-test is: $df = n - 1 = 13 - 1 = 12$

---

Step 4: Compute the P-Value

Using a TI-84 calculator:

1. Press 2nd → DISTR → tcdf(.
2. Input the bounds for the t-distribution:
   • Lower bound = $-\infty$ (use a very large negative value like $-100$),
   • Upper bound = $-1.37$ (for the left tail),
   • Degrees of freedom = 12.
3. Repeat for the upper tail with $t = 1.37$.
4. Multiply the result by 2 (since it’s a two-tailed test).

Alternatively, approximate manually: Using t-distribution tables, for $t = 1.37$ and $df = 12$, the one-tailed P-value is approximately 0.1. Thus, the two-tailed P-value is: $P \approx 2 \times 0.1 = 0.2$

---

Step 5: Decision Rule

At $\alpha = 0.01$:

• Compare the P-value (0.2) to $\alpha$.
• Since $P > \alpha$, fail to reject the null hypothesis.

---

Conclusion

There is insufficient evidence at the $\alpha = 0.01$ level of significance to conclude that the mean tuition and fees for private colleges in California differs from $35,000.

---

Would you like further clarification or step-by-step instructions for using the TI-84?

---

5 Questions to Explore Further

1. How is the t-distribution different from the standard normal distribution?
2. What happens if the sample size increases while keeping all other variables constant?
3. How do we determine whether to use a one-tailed or two-tailed test?
4. What are the assumptions of the single-sample t-test?
5. How would the result change if $\alpha = 0.05$ instead of 0.01?

Tip:

Always check if the population is approximately normal or if the sample size is large ($n \geq 30$) before using t-tests to ensure valid results.