The equation of the regression line for the paired data below is ŷ = 3x and the standard error estimate is SE = 2.2361. Find the 90% prediction interval of y for x = 3.
x = [2, 4, 5, 6]
y = [7, 11, 13, 20]

Which is the correct answer?
Answer A. 6.8 < y < 11.2
Answer B. 4.5 < y < 13.5
Answer C. 7.1 < y < 10.9
Answer D. 1.2 < y < 16.8

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.

Learn how to calculate a prediction interval in regression analysis using the regression equation and standard error estimate. Solve a specific problem with paired data and determine the correct prediction interval. Suitable for advanced high school and college students.

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