12. positive events are great, but recent research suggests that unexpected positive outcomes (e.g., an…

12. positive events are great, but recent research suggests that unexpected positive outcomes (e.g., an unseasonably sunny day) predict greater - than - normal amounts of risk - taking and gambling (otto, fleming, & glimcher, 2016). researchers demonstrated this by comparing lottery sales—indicative of risk - taking—on normal days with lottery sales on days when some unexpected positive event occurred in the city. they observed increased sales after unexpected positive outcomes. suppose that a researcher extends this observation to the laboratory and randomly assigns participants to two groups. group 1 receives an unexpectedly large payment for participating and group 2 receives the expected amount of compensation. the researcher then measures how much money the participants are willing to gamble in a game of chance.\n\n\n\ntest the one - tailed hypothesis that an unexpected positive outcome increased the amount of money that participants were willing to gamble. use $\\alpha=.01$.

12. positive events are great, but recent research suggests that unexpected positive outcomes (e.g., an unseasonably sunny day) predict greater - than - normal amounts of risk - taking and gambling (otto, fleming, & glimcher, 2016). researchers demonstrated this by comparing lottery sales—indicative of risk - taking—on normal days with lottery sales on days when some unexpected positive event occurred in the city. they observed increased sales after unexpected positive outcomes. suppose that a researcher extends this observation to the laboratory and randomly assigns participants to two groups. group 1 receives an unexpectedly large payment for participating and group 2 receives the expected amount of compensation. the researcher then measures how much money the participants are willing to gamble in a game of chance.\n\n\n\ntest the one - tailed hypothesis that an unexpected positive outcome increased the amount of money that participants were willing to gamble. use $\\alpha=.01$.

Answer

Explanation:

Step1: Calculate the pooled variance

The formula for pooled variance (s_{p}^{2}=\frac{SS_{1} + SS_{2}}{df_{1}+df_{2}}). Given (n_{1}=n_{2} = 16), so (df_{1}=n_{1}-1=15), (df_{2}=n_{2}-1 = 15), (SS_{1}=6.5), (SS_{2}=10.0). (s_{p}^{2}=\frac{6.5 + 10.0}{15+15}=\frac{16.5}{30}=0.55)

Step2: Calculate the standard error

The formula for standard error (s_{M_{1}-M_{2}}=\sqrt{\frac{s_{p}^{2}}{n_{1}}+\frac{s_{p}^{2}}{n_{2}}}) Since (n_{1}=n_{2} = 16) and (s_{p}^{2}=0.55) (s_{M_{1}-M_{2}}=\sqrt{\frac{0.55}{16}+\frac{0.55}{16}}=\sqrt{\frac{0.55\times2}{16}}=\sqrt{\frac{1.1}{16}}\approx\sqrt{0.06875}\approx0.262)

Step3: Calculate the t - statistic

The formula for t - statistic (t=\frac{(M_{1}-M_{2})-( \mu_{1}-\mu_{2})}{s_{M_{1}-M_{2}}}) Assume (\mu_{1}-\mu_{2} = 0) (null hypothesis), (M_{1}=5.75), (M_{2}=5.00) (t=\frac{(5.75 - 5.00)-0}{0.262}=\frac{0.75}{0.262}\approx2.86)

Step4: Determine the critical t - value

For a one - tailed test with (\alpha = 0.01) and (df=df_{1}+df_{2}=30) Looking up the t - distribution table, the critical (t) value (t_{crit}) (one - tailed, (\alpha = 0.01), (df = 30)) is (t_{crit}= 2.457)

Step5: Make a decision

Since the calculated (t = 2.86>t_{crit}=2.457)

Answer:

We reject the null hypothesis. There is sufficient evidence at the (\alpha = 0.01) level to support the one - tailed hypothesis that an unexpected positive outcome increased the amount of money that participants were willing to gamble.