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To compute a 95% confidence interval for the mean number of visits for physical therapy patients, let's go step-by-step:

### Step 1: Identify the correct distribution
Since the sample size is small (n = 12), and we do not know the population standard deviation, we will use the **t-distribution** (rather than the normal distribution). So, for part (a), the answer is **t-distribution**.

### Step 2: Calculate the sample mean and standard deviation
Given the dataTo compute a 95% confidence interval for the mean number of visits for physical therapy patients, let's go step-by-step:

### Step 1: Identify the correct distribution
Since the sample size is small (n = 12), and we do not know the population standard deviation, we will use the **t-distribution** (rather than the normal distribution). So, for part (a), the answer is **t-distribution**.

### Step 2: Calculate the sample mean and standard deviation
Given the 
\[
21, 25, 18, 9, 26, 24, 9, 12, 23, 5, 25, 10
\]
- Sample size \( n = 12 \)
- To find the sample mean \( \bar{x} \) and sample standard deviation \( s \), we will perform the following calculations:

#### Mean calculation:
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{21 + 25 + 18 + 9 + 26 + 24 + 9 + 12 + 23 + 5 + 25 + 10}{12}
\]
\[
\bar{x} = \frac{207}{12} = 17.25
\]

#### Standard deviation calculation:
To calculate the sample standard deviation \( s \), we use:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]
The deviations from the mean for each value are calculated, then squared, summed, and divided by \( n-1 \), followed by taking the square root.

\[
s = \sqrt{\frac{(21-17.25)^2 + (25-17.25)^2 + ... + (10-17.25)^2}{11}} \approx 7.044
\]

### Step 3: Compute the margin of error
For a 95% confidence interval, we need the critical value of the **t-distribution** for \( n - 1 = 11 \) degrees of freedom, which can be found in a t-table or calculator. The value of \( t_{0.025, 11} \) is approximately **2.201**.

The margin of error (E) is calculated as:
\[
E = t \cdot \frac{s}{\sqrt{n}} = 2.201 \cdot \frac{7.044}{\sqrt{12}} = 4.473
\]

### Step 4: Calculate the confidence interval
Now, to find the confidence interval for the mean:
\[
\text{Lower bound} = \bar{x} - E = 17.25 - 4.473 = 12.777
\]
\[
\text{Upper bound} = \bar{x} + E = 17.25 + 4.473 = 21.723
\]

So, for part (b), with 95% confidence, the population mean number of visits is between **12.777** and **21.723**.

### Step 5: Interpretation of the confidence interval (part c)
For part (c), if many groups of 12 patients are studied, about **95** percent of these confidence intervals will contain the true population mean, while **5** percent will not.

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Let me know if you would like more details on any step, or if you have any questions!

Here are 5 related questions:
1. How does the sample size affect the width of the confidence interval?
2. Why do we use the t-distribution instead of the normal distribution for small samples?
3. How does increasing the confidence level (e.g., to 99%) affect the confidence interval?
4. What assumptions must be met to use the t-distribution for confidence intervals?
5. How would the confidence interval change if the population standard deviation was known?

**Tip**: Always check if the sample size is large enough or if you know the population standard deviation before deciding on the type of distribution to use.

You are interested in finding a 95% confidence interval for the mean number of visits for physical therapy patients. The data below show the number of visits for 12 randomly selected physical therapy patients. Round answers to 3 decimal places where possible.

Calculate the 95% confidence interval for the mean number of visits for 12 randomly selected physical therapy patients using the t-distribution. Learn how to find the sample mean, standard deviation, and margin of error for this statistics problem.

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