Answer
$t_0\lt t_α$: null hypothesis is not rejected.
There is not enough evidence to conclude that people spend more when using a credit card.
Work Step by Step
- Mean;
- Independent sampling.
$x ̅_1,n_1~and~s_1$ refer to credit card and $x ̅_2,n_2~and~s_2$ refer to cash.
$x ̅_1=\frac{∑x_{1_i}}{n_1}=16.972$
$s_1=\sqrt {\frac{∑(x_{1_i}-x ̅_1)^2}{n_1}}=3.989$
$x ̅_2=\frac{∑x_{2_i}}{n_2}=12.343$
$s_2=\sqrt {\frac{∑(x_{2_i}-x ̅_2)^2}{n_2}}=5.151$
$H_0:~µ_1=µ_2$ versus $H_1:~µ_1\gt µ_2$
$t_0=\frac{(x ̅_1-x ̅_2)-(µ_1-µ_2)}{\sqrt {\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}=\frac{(16.972-12.343)-0}{\sqrt {\frac{3.989^2}{10}+\frac{5.151^2}{10}}}=2.247$
$n=10$, so:
$d.f.=n-1=9$
Right-tailed test:
$t_α=t_{0.01}=2.821$
(According to Table VI, for d.f. = 9 and area in right tail = 0.01)
Since $t_0\lt t_α$, we do not reject the null hypothesis.
