To construct a 90% confidence interval for the mean tuition of all colleges and universities in the United States, we'll follow these steps:

Step 1: Identify the given data

• Sample mean ($\bar{x}$) = $19,200
• Sample standard deviation ($s$) = $11,000
• Sample size ($n$) = 40
• Confidence level = 90%

Step 2: Find the t-value

Since the sample size is less than 30, we'll use the t-distribution. For a 90% confidence interval, we want the t-value that corresponds to $\frac{90\%}{2} = 0.05$ in each tail of the distribution. The degrees of freedom (df) = $n - 1 = 40 - 1 = 39$.

From the t-table, the critical t-value for a 90% confidence interval with 39 degrees of freedom is approximately 1.685.

Step 3: Calculate the standard error of the mean (SE)

$SE = \frac{s}{\sqrt{n}} = \frac{11,000}{\sqrt{40}} = \frac{11,000}{6.32} \approx 1,740.51$

Step 4: Calculate the margin of error (ME)

$ME = t \times SE = 1.685 \times 1,740.51 \approx 2,931.96$

Step 5: Construct the confidence interval

The confidence interval is given by: $\text{Confidence Interval} = \bar{x} \pm ME = 19,200 \pm 2,931.96$ $\text{Lower bound} = 19,200 - 2,931.96 = 16,268.04$ $\text{Upper bound} = 19,200 + 2,931.96 = 22,131.96$

Final Confidence Interval:

$\boxed{(16,268.04, 22,131.96)}$ Thus, we are 90% confident that the mean tuition for all colleges and universities in the United States falls between $16,268.04 and $22,131.96.

Would you like more details on any part of this explanation?

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Here are five related questions:

1. How would the confidence interval change if the sample size were larger?
2. What happens to the confidence interval if we switch to a 95% confidence level?
3. Why do we use the t-distribution for this problem instead of the normal distribution?
4. How does the standard deviation affect the width of the confidence interval?
5. If the standard deviation were smaller, how would that influence the confidence interval?

Tip: Increasing the sample size reduces the standard error, which in turn narrows the confidence interval, leading to more precise estimates.