To solve this problem, we need to calculate the 90% prediction interval for $y$ when $x = 3$ using the regression line equation $\hat{y} = 3x$ and the provided standard error of estimate $SE = 2.2361$.

Step 1: Calculate the predicted value of $y$ when $x = 3$

Using the regression equation $\hat{y} = 3x$:

$\hat{y} = 3(3) = 9$

Step 2: Determine the critical value for a 90% prediction interval

Since we are dealing with a 90% prediction interval, we look for the critical value $t_{\alpha/2}$ for a 90% confidence level. For a small sample size, this value is typically found using the t-distribution table with $n-2$ degrees of freedom. Here, $n = 4$, so the degrees of freedom $df = 4 - 2 = 2$.

From the t-distribution table, the critical value $t_{\alpha/2}$ for $df = 2$ and a 90% confidence level is approximately $t_{0.05, 2} \approx 2.920$.

Step 3: Calculate the margin of error

The margin of error for the prediction interval is calculated as follows:

$\text{Margin of Error} = t_{\alpha/2} \times SE = 2.920 \times 2.2361 \approx 6.528$

Step 4: Construct the prediction interval

The prediction interval for $y$ when $x = 3$ is:

$\hat{y} \pm \text{Margin of Error} = 9 \pm 6.528$

This gives us:

$9 - 6.528 < y < 9 + 6.528$

$2.472 < y < 15.528$

Step 5: Compare with the provided answers

The closest range to our calculated interval is:

• Answer D: $1.2 < y < 16.8$

This interval encompasses the calculated prediction interval $2.472 < y < 15.528$.

Final Answer

The correct answer is Answer D: $1.2 < y < 16.8$.

Would you like more details or have any questions? Here are five related questions for further exploration:

1. How do we calculate the standard error of estimate in regression analysis?
2. What is the difference between a confidence interval and a prediction interval?
3. Why do we use the t-distribution instead of the z-distribution for small samples?
4. How would the prediction interval change if the standard error were smaller?
5. How do outliers in the data affect the prediction interval?

Tip: Always check the sample size before deciding whether to use the t-distribution or the z-distribution for constructing intervals.