Answer to Question #115882 in Statistics and Probability for Shane Jessica Ballera

Question

Hypothesis Testing

3. Your friend has been betting at an average $25 in Las Vegas for the last 15 games which differs by around $10. You can accuse him of being overly risky when you learned that clients bet only $20 per game. At 95% level of certainty, are you correct?

Answer 1

$a_0=20, \overline{x}=25, s=10, n=15, \alpha=0.05\\ H_0: \mu=a_0=20, H_1: \mu>a_0=20\text{ (right-tailed)}$

We assume that bet has normal distribution.

We will use the following random variable as a criterion:

$U=\frac{(\overline{X}-a_0)\sqrt{n}}{s}$

This random variable has t-distribution with $k=n-1$ degrees of freedom.

Observed value:

$u_{obs}=\frac{(25-20)\sqrt{15}}{10}\approx 1.94$

Critical value (one-sided):

$t_{cr}=t_{cr}(\alpha;k)=t_{cr}(0.05;14)\approx 1.76$

$(1.76;\infty)\text{ --- critical region}$

$u_{obs}$ falls into the critical region. So we reject $H_0$.

We can accuse him of being overly risky (if clients bet only $20 per game).