Part 1: Practice with the Big Ideas Have you ever played the game Rock-PaperScissors? Rock-PaperScissors is a two-player game that is often used to quickly determine a winner. In the game, each player puts out a st (rock), a at hand (paper), or a hand with two ngers extended (scissors). For those who might be unfamiliar with this game, the diagram below shows these three hand congurations. In this game, rock beats scissors, scissors heats paper, and paper beats rock. A player clearly has three options to choose omrock, paper, or scissorswhen making the rst move in the game. Are the three options selected equally often? This is important to know, because knowing how frequently options are chosen would give a player a signicant advantage in the game. @5535 BUCK SGISSIJHS PAPER Suppose it is claimed that when playing the game Rock-PaperScissors, choosing \"paper\" as the rst move in the game is just as likely as choosing \"rock\" or \"scissors.\" In other words, in the population, the proportion of game players who will choose \"paper\" as a rst move is 0.333. A researcher questions this claim. He believes that choosing \"paper\" as a rst move in the game happens less often than what is claimed. The researcher observes a random sample of 200 people playing Rock-Paper-Scissors and nds that 62 of these people choose \"paper\" as a rst move in the game. Use this information to answer Questions 1 through 7. To begin, we need to remember that with hypothesis testing, we always start with a set of two competing hypotheses. We have a hypothesis that puts forward the claim that is being tested about the population. We call this hypothesis the null hypothesis. We then have a hypothesis that illustrates what we think is really going on in the population. We call this hypothesis the alternative hypothesis. Using appropriate symbols, our hypotheses in this example are as follows: Ho: ,0 = 0.333 Ha: p \" Sign or a \"7E\" sign? 3. Given that 62 out of a random sample of 200 players chose \"paper\" as a rst move, what will the sample proportion (or 15) be? Please compute this value below. 4. To be able to conduct a hypothesis test, we will now need to compute a test statistic (using the following formula). Please attempt to compute this test statistic below, showing as much work as you can. To be as precise as you can be, round the denominator to three decimal places before dividing the value from the numerator by the denominator. A pip p(1 - p) T! z: As part of the process of conducting a hypothesis test, we need to nd what's called a probability value, or a Pvalue for short. This Pvalue value tells us something about how likely it would be to observe a sample outcome as extreme or more extreme than what we observed, if the null hypothesis is reallytrug. The smaller the P-value, the more unlikely the sample outcome would be under the assumption that the null hypothesis is true. We determine if we have evidence against the null hypothesis, in favor of the alternative hypothesis, based on how the P-value compares to our chosen signicance (or alpha) level. If the P-value is less than or equal to the signicance level, we say that we have evidence against the null hypothesis. If the Pvalue is larger than the signicance level, we do not have enough evidence against the null hypothesis (i.e., we cannot rule out the null hypothesis as a reasonable explanation for the observed sample outcome). 5. Based on the test statistic you calculated to answer Question 4, what is the P-value equal to? Use Table B to nd this Pvalue. (Hint: Don't forget that since a Pvalue is a probability, you will now need to take the percentile you get from Table B and divide it by 100 to convert it to a probability). 6. Assume here that we are using a signicance level of 0.05. Based on this information, along with the Pvalue you obtained in your answer to Question 5, what should we conclude about the null hypothesis, and why? Let's now use a technology tool to verify some of your work and to show you a way to better visualize this problem. We can use an applet from the Art of Stat site to conduct a hypothesis on condence intervals. Please open the applet by clicking on the link below. https://istats.shinyappsio/Inference prop/ Once you open the applet, go to where it says \"Type of Inference\" in the middle part of the left side of the page, and select Signicance Test. In the upper left comer, where it $31; \"Enter Data,\" use the dropdown menu to select Number of Successes. There, you will type in a Sample Size and then the # of Successes. Here, for our Rock-Paper-Scissors example, you'll want to type in a sample size of 200 and a_ _r_11_1_r_r_il_3_er_9_f successes equal to 62. Once you type in these numbers, you'll see some output appear on the right side of screen. You'll then want to type in a \"Null Value\" (i.e., the value of p), and select the appropriate direction for the Alternative hypothesis (i.e., Less for this example, since we had the \"