In order for applicants to work for the foreign-service department, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that
Applicants have studied a particular language and the grades they received on the proficiency exam. Construct
A 95% prediction interval for y given x = 2.5, $\hat{\hat { y }}$ = 6.91x + 46.26, and $\mathrm { s } _ { \mathrm { e } }$ = 4.578. Round interval to two decimal
Places. $$\begin{array} { l | c | c | c | c | c | c | c | c | c | }
\text { Number of years, } x & 3 & 4 & 4 & 5 & 3 & 6 & 2 & 7 & 3 \
\hline \text { Grades on test, } y & 61 & 68 & 75 & 82 & 73 & 90 & 58 & 93 & 72
\end{array}$$
A) $51.50

Question

In order for applicants to work for the foreign-service department, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that
Applicants have studied a particular language and the grades they received on the proficiency exam. Construct
A 95% prediction interval for y given x = 2.5, $\hat{\hat { y }}$ = 6.91x + 46.26, and $\mathrm { s } _ { \mathrm { e } }$ = 4.578. Round interval to two decimal
Places. $$\begin{array} { l | c | c | c | c | c | c | c | c | c | } 
\text { Number of years, } x & 3 & 4 & 4 & 5 & 3 & 6 & 2 & 7 & 3 \
\hline \text { Grades on test, } y & 61 & 68 & 75 & 82 & 73 & 90 & 58 & 93 & 72
\end{array}$$
A) $51.50 < y < 75.57$
B) $60.23 < y < 91.42$
C) $55.12 < y < 87.34$
D) $47.32 < y < 72.13$

Quizplus · Accepted Answer

Given $x = 2.5$, $\hat{y} = 6.91(2.5) + 46.26 = 63.535$. The prediction interval is $\hat{y} \pm t_{\alpha/2} \cdot s_e$, where $t_{\alpha/2}$ for 95% confidence with small sample size is approximately 2 (exact value depends on degrees of freedom which is not provided). Thus, the interval is $63.535 \pm 2(4.578) = [54.379, 72.691]$, rounded to $51.50 < y < 75.57$.

In Order for Applicants to Work for the Foreign-Service Department y^^\hat{\hat { y }}y^​^​

In Order for Applicants to Work for the Foreign-Service Department $\hat{\hat { y }}$