Question

1.5.17 Have you ever played rock-paper-scissors (or Rochambeau)? It's considered a "fair game" in that the two players are equally likely to win (like a coin toss). Both players simultaneously display one of three hand gestures (rock, paper, or scissors), and the objective is to display a gesture that defeats that of your opponent. The main gist is that rocks break scissors, scissors cut paper, and paper covers rock. We investigated some results of the game rock-paper-scissors, where the researchers had 119 people play rock-paper-scissors against a computer. They found 66 players (55.5%) started with rock, 39 (32.8%) started with paper, and 14 (11.8%) started with scissors. We want to see if players start with scissors with a long-term probability that is different from 1/3. Which of the following states the appropriate null and alternative hypotheses in the context of this study, first in words and then in symbols? 0 Null: The long-run proportion of times that a player starts with scissors is 33%, -33% Alt: The long-run proportion of times that a player starts with scissors is different from 33%, п 33% 0 Null: The long-run proportion of times that a player starts with scissors is 50%, -50% Alt: The long-run proportion of times that a player starts with scissors is different from 50%, π 50% Null: The long-run proportion of times that a player starts with scissors is 33%, -33% Alt: The long-run proportion of times that a player starts with scissors is more than 33%, π-33% Null: The long-run proportion of times that a player starts with scissors is 33%, -33% Alt: The long-run proportion of times that a player starts with scissors is less than 33%, π < 33%

please give answers in a,b,c,d format and give calculation.

Answer 1

Sample size, n = 119

Number of people that start with scissors, x = 14

Sample proportion, $\bar p$ = 14/119 = 0.118

Null and Alternative hypothesis:

Null: The long-run proportion of times that a player starts with scissors is 33%, $\pi$ = 33%

Alt: The long-run proportion of times that a player starts with scissors is different from 33%, $\pi$$\ne$ 33%

Test statistic:

$z = \frac{\bar p - p_0}{\sqrt{p_0(1-p_0)/n}} = \frac{ 0.118 - 0.33 }{\sqrt{ 0.33(1- 0.33)/119}} = \textbf{-4.926}$

P-value:

p-value = 0.00

Conclusion:

As p = 0 < $\alpha$, it is concluded that the null hypothesis is rejected.

We have very strong evidence that the long run proportion of times that a player starts with scissors is different from 33%