For a sample of size 15 , the following values were obtained: $b _ { 0 } = 3.58 , b _ { 1 } = 4.79 , s _ { \mathrm { e } } = 4.09$, $\sum ( x - \bar { x } ) ^ { 2 } = 237.89$, and $\bar { x } = 3.00$
Construct a $95 \%$ confidence interval for the mean response when $x = 0$.
A) (0.59, 6.57)
B) (1.24, 5.92)
C) (0.72, 6.44)
D) (0, 5.92)

Question

Quizplus · Accepted Answer

The formula for the confidence interval for the mean response when x = 0 is:
$$\hat { y } _ { 0 } \pm t _ { \frac { \alpha } { 2 } , n - 2 } s _ { e } \sqrt { 1 + \frac { 1 } { n } + \frac { ( 0 - \bar { x } ) ^ { 2 } } { \sum ( x - \bar { x } ) ^ { 2 } } }$$

Plugging in the given values, we get:
$$\hat { y } _ { 0 } \pm t _ { \frac { 0.05 } { 2 } , 13 } (4.09) \sqrt { 1 + \frac { 1 } { 15 } + \frac { ( 0 - 3 ) ^ { 2 } } { 237.89 } }$$

Using a t-table, the value of $t _ { \frac { 0.05 } { 2 } , 13 } = 2.160$

Plugging this back into the formula and simplifying, we get:
$$\hat { y } _ { 0 } \pm (2.160)(4.09) (0.692)$$
$$\hat { y } _ { 0 } \pm 6.23$$

Therefore, the 95% confidence interval for the mean response when $x = 0$ is:
$$\hat { y } _ { 0 } \pm 6.23 \rightarrow 3.58 \pm 6.23 \rightarrow (0.72, 6.44)$$

So the correct answer is C.

For a Sample of Size 15 , the Following Values b0=3.58,b1=4.79,se=4.09b _ { 0 } = 3.58 , b _ { 1 } = 4.79 , s _ { \mathrm { e } } = 4.09b0​=3.58,b1​=4.79,se​=4.09

For a Sample of Size 15 , the Following Values $b _ { 0 } = 3.58 , b _ { 1 } = 4.79 , s _ { \mathrm { e } } = 4.09$