Answer
$t_0\lt -t_α$: null hypothesis is rejected.
There is enough evidence to conclude that sleep disorders adversely affect one’s GPA.
Work Step by Step
$x ̅_1,n_1~and~s_1$ refer to students with a sleep disorder and $x ̅_2,n_2~and~s_2$ refer to students without a sleep disorder.
$H_0:~µ_1=µ_2$ versus $H_1:~µ_1\lt µ_2$
$t_0=\frac{(x ̅_1-x ̅_2)-(µ_1-µ_2)}{\sqrt {\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}=\frac{(2.65-2.82)-0}{\sqrt {\frac{0.87^2}{503}+\frac{0.83^2}{1342}}}=-3.784$
$n=503$ (use the smaller value of $n$), so:
$d.f.=n-1=502$
Left-tailed test:
$t_α=t_{0.05}=1.660$
(According to Table VI, for d.f. = 100, the closest value to 502, and area in right tail = 0.05)
So, $-t_α=-1.660$
Since $t_0\lt -t_α$, we reject the null hypothesis.
