A journal article reported that the mean hospital stay following a particular surgical procedure in 2001 was 7.1 days. A researcher feels that the mean hospital stay in 2002 should be less due to initiatives aimed at reducing health care costs. A random sample of 40 patients undergoing the same surgical procedure in 2002 had a mean length of stay of 6.85 days with a standard deviation of 7.01 days. Run the appropriate statistical test at α=0.05. 
Choose the correct hypotheses.



a.  H0:μ=7.1   H1:μ<7.1 



b.  H0:μ=7.1   H1:μ≠7.1 



c.  H0:μ=7.1   H1:μ<6.85 



d.  H0:μ=7.1   H1:μ≠6.85 



e.
H0:μ=7.1   H1:μ>7.1.

The goal of this test is to determine whether the mean hospital stay in 2002 is less than it was in 2001, based on a sample of 40 patients. This is a **one-tailed hypothesis test** since the researcher believes the mean has decreased.

To set up the hypotheses:

- The null hypothesis (\(H_0\)) states that the mean hospital stay has not decreased and is equal to 7.1 days, the mean from 2001.
- The alternative hypothesis (\(H_1\)) expresses the researcher's belief that the mean hospital stay in 2002 is less than 7.1 days.

Thus, the correct hypotheses should be:

- \(H_0: \mu = 7.1\) (the mean stay is still 7.1 days)
- \(H_1: \mu < 7.1\) (the mean stay is less than 7.1 days)

This matches **option a**:

**a.  \(H_0: \mu = 7.1\), \(H_1: \mu < 7.1\)**.

Would you like me to walk through the steps of performing the statistical test?

Here are 5 related questions:
1. How do you calculate the test statistic for this hypothesis test?
2. What is the critical value for a one-tailed test at \(\alpha = 0.05\)?
3. What assumptions must be met for this test to be valid?
4. How would you interpret the p-value in the context of this test?
5. What would change if this were a two-tailed test?

**Tip**: Always ensure the sample size is large enough when using the t-test, especially for small samples. For large samples (n > 30), the normal approximation is often acceptable.

A journal article reported that the mean hospital stay following a particular surgical procedure in 2001 was 7.1 days. A researcher feels that the mean hospital stay in 2002 should be less due to initiatives aimed at reducing health care costs. A random sample of 40 patients undergoing the same surgical procedure in 2002 had a mean length of stay of 6.85 days with a standard deviation of 7.01 days. Run the appropriate statistical test at α=0.05. Choose the correct hypotheses: a. H0:μ=7.1 H1:μ<7.1; b. H0:μ=7.1 H1:μ≠7.1; c. H0:μ=7.1 H1:μ<6.85; d. H0:μ=7.1 H1:μ≠6.85; e. H0:μ=7.1 H1:μ>7.1.

Analyze a hypothesis test to determine if the mean hospital stay in 2002 is less than in 2001 based on a sample of 40 patients. This problem involves calculating the t-statistic and p-value for a one-tailed test at α=0.05, utilizing the t-distribution and central limit theorem.

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