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This problem involves hypothesis testing for a mean, where the sample data is provided, and you're asked to find the p-value. Let's break down the steps to solve it:

### Step 1: State the Hypotheses
- **Null Hypothesis (H₀):** \(\mu = 7\) (the average number of empty beds is 7).
- **Alternative Hypothesis (H₁):** \(\mu \neq 7\) (the average number of empty beds is not 7).

### Step 2: Find the Sample Mean (\(\bar{x}\)) and Sample Standard Deviation (s)
We are given the data points. We need to compute the sample mean and sample standard deviation.

The data points are:
\[ 10, 3, 12, 3, 12, 2, 7, 0, 14, 5, 3, 11, 14, 15, 18, 13, 3, 10, 3, 2, 7, 10, 12, 12, 13, 7, 13, 8, 7, 10 \]

### Step 3: Calculate the Test Statistic
The test statistic for a one-sample t-test is:
\[
t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]
Where:
- \(\bar{x}\) is the sample mean
- \(\mu_0 = 7\) is the hypothesized population mean
- \(s\) is the sample standard deviation
- \(n\) is the sample size

### Step 4: Find the p-value
Once we compute the t-statistic, we'll find the corresponding p-value from the t-distribution with \(n - 1\) degrees of freedom.

### Step 5: Conclusion
We'll compare the p-value to \(\alpha = 0.10\). If the p-value is less than \(\alpha\), we reject the null hypothesis.

Let me compute the sample statistics (mean and standard deviation) and calculate the t-statistic.### Results:
- **Sample Mean**: \( \bar{x} = 8.63 \)
- **Sample Standard Deviation**: \( s = 4.72 \)
- **Test Statistic**: \( t = 1.89 \)
- **p-value**: \( p = 0.068 \)

### Conclusion:
Since the p-value is \(0.068\), which is less than \(\alpha = 0.10\), we reject the null hypothesis. This suggests that there is significant evidence to believe that the average number of empty beds is different from 7.

Do you have any questions or want further details on any part of the process?

Here are 5 related questions to expand your understanding:
1. How is the test statistic calculated for a one-sample t-test?
2. What does the p-value represent in hypothesis testing?
3. How do we interpret the result of a hypothesis test at different significance levels (\(\alpha\))?
4. What is the difference between a one-tailed and two-tailed test?
5. How would the result change if the sample size were larger or smaller?

**Tip**: Always double-check if you're conducting a one-tailed or two-tailed test based on the alternative hypothesis.

The hospital has felt emptier this year for some reason. To determine if this is true, you decide to collect data and evaluate it using α = 0.10. Given the generated sample data provided and a null hypothesized value of H₀: μ = 7 empty beds on average, what is the p-value? The sample data is: 10, 3, 12, 3, 12, 2, 7, 0, 14, 5, 3, 11, 14, 15, 18, 13, 3, 10, 3, 2, 7, 10, 12, 12, 13, 7, 13, 8, 7, 10.

Learn how to perform hypothesis testing on the average number of empty hospital beds using t-distribution. The problem includes sample data, significance level (α = 0.10), and calculates the p-value for determining if the average is different from 7.

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