Let's go through each part step-by-step.

Part (a): Distribution Type

Since the sample size is relatively small (11), and we are not given the population standard deviation, we should use a t-distribution to calculate the confidence interval. This is because the t-distribution is appropriate when working with small samples and an unknown population standard deviation.

Part (b): Calculating the 90% Confidence Interval

1. Find the sample mean ($\bar{x}$) and sample standard deviation (s):

   • Data: $27, 28, 13, 23, 10, 5, 14, 25, 22, 24, 28$

   Let's calculate the mean and standard deviation:

   $\bar{x} = \frac{27 + 28 + 13 + 23 + 10 + 5 + 14 + 25 + 22 + 24 + 28}{11}$

   After calculating, the mean ($\bar{x}$) is approximately $19.636$ (rounded to 3 decimal places).

   Now, we calculate the sample standard deviation (s).

2. Determine the t-score for a 90% confidence level with 10 degrees of freedom (df = n - 1 = 11 - 1):

   • From the t-distribution table, the t-score for a 90% confidence interval with 10 degrees of freedom is approximately 1.812.

3. Calculate the margin of error (ME): $ME = t \times \frac{s}{\sqrt{n}}$

4. Calculate the confidence interval: $\text{CI} = \bar{x} \pm ME$

I'll compute these values for the precise interval shortly.

Part (c): Interpretation of the Confidence Interval

A 90% confidence level means that if we were to take many random samples of 11 patients and calculate a confidence interval from each sample, about 90% of those intervals would contain the true population mean, while about 10% would not.

Let me finish these calculations and share the exact interval with you.Here are the calculated values:

• Sample Mean ($\bar{x}$): 19.909 (rounded to 3 decimal places)
• Sample Standard Deviation (s): 8.006
• t-Score for 90% Confidence (df = 10): 1.812
• Margin of Error (ME): 4.375

Answers to the Parts:

1. Part (a): Use a t-distribution.
2. Part (b): With 90% confidence, the population mean number of visits per physical therapy patient is between 15.534 and 24.284 visits.
3. Part (c): About 90% of the confidence intervals will contain the true population mean, and about 10% will not contain the true population mean.

Would you like further details on any specific part or have additional questions?

Further Questions:

1. How does the sample size affect the choice between a t-distribution and a z-distribution?
2. What changes if the confidence level is increased to 95%?
3. How does the margin of error impact the width of the confidence interval?
4. What happens to the confidence interval if more data points are added to the sample?
5. How is the confidence interval interpreted in real-world research studies?

Tip:

When dealing with small sample sizes and unknown population standard deviations, the t-distribution provides more accurate confidence intervals than the z-distribution.