: The Fair Isaac Corporation (FICO) credit score is used by banks and other lenders to determine whether someone is a good credit risk. Scores range from 300  to 850 , with a score of  720 or more indicating that a person is a very good credit risk. An economist wants to determine whether the mean FICO score is lower than the cutoff of 720.  She finds that a random sample of 60 people had a mean FICO score of  695 with a standard deviation of 65 . Can the economist conclude that the mean FICO score is less than 720 ? Use the a=0.05  level of significance and the P-value method with the TI-84 Plus calculator.

To determine whether the mean FICO score is lower than the cutoff of 720, we'll conduct a hypothesis test for the mean using the sample data provided. We'll use a \( \alpha = 0.05 \) level of significance and the P-value method. Here are the steps:

1. **Define the hypotheses:**

   - Null hypothesis (\(H_0\)): \( \mu = 720 \)
   - Alternative hypothesis (\(H_1\)): \( \mu < 720 \)

2. **Calculate the test statistic (t):**

   The test statistic for a sample mean when the population standard deviation is unknown is calculated using the t-distribution:

   \[
   t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
   \]

   Given:
   - Sample mean (\(\bar{x}\)) = 695
   - Hypothesized population mean (\(\mu_0\)) = 720
   - Sample standard deviation (s) = 65
   - Sample size (n) = 60

   \[
   t = \frac{695 - 720}{65 / \sqrt{60}} \approx \frac{-25}{8.385} \approx -2.98
   \]

3. **Determine the degrees of freedom (df):**

   \[
   df = n - 1 = 60 - 1 = 59
   \]

4. **Find the P-value:**

   Using the TI-84 Plus calculator, we can find the P-value for the t-test.

   - Press [2ND] then [VARS] to access the DISTR menu.
   - Select "tcdf(".
   - Enter the lower bound as \(-9999\) (a very large negative number), the upper bound as \(-2.98\), and the degrees of freedom as 59.

   So, the input will be `tcdf(-9999, -2.98, 59)`.

   The calculator will return the P-value. For our calculations:

   \[
   \text{P-value} \approx 0.0020
   \]

5. **Compare the P-value with \( \alpha \):**

   \[
   \text{P-value} = 0.0020 < 0.05 = \alpha
   \]

Since the P-value is less than the significance level \( \alpha \), we reject the null hypothesis.

6. **Conclusion:**

   There is sufficient evidence at the \( \alpha = 0.05 \) level of significance to conclude that the mean FICO score is less than 720.

---

Would you like more details or have any questions?

Here are 8 related questions for further exploration:

1. How would the conclusion change if the level of significance was \( \alpha = 0.01 \)?
2. What if the sample size was larger, say 100 people? How would that affect the test?
3. How does changing the hypothesized mean (\( \mu_0 \)) to 710 affect the test?
4. What are Type I and Type II errors in the context of this hypothesis test?
5. How would you calculate the confidence interval for the mean FICO score?
6. How can you use the TI-84 Plus to conduct a two-sample t-test?
7. What assumptions must be satisfied for this hypothesis test to be valid?
8. How does the choice of the level of significance (\( \alpha \)) influence the hypothesis test results?

**Tip:** Always ensure the sample size is sufficiently large to approximate the t-distribution to the normal distribution, especially when the population standard deviation is unknown.

Explore the analysis of mean FICO scores using hypothesis testing. Learn how to determine if a mean FICO score is lower than a specified cutoff using the P-value method and a sample size of 60. Detailed step-by-step solution with a significance level of 0.05.

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