Please help me understand why the chi square statistic is 6.21 for part d and also the degrees of freedom. Thank you. (The solution for part d is attached.) No need to explain anything else, just part d please. (How do I get 6.21 by calculation)

Problem 2 Due to the global crisis, the 2020 Rock Paper Scissors competition was moved online. Lucky fer you, this allowed you to record each game and tally up the average choices of Rock, Paper, and Scissors. If you could nd a disproportionate mimber of one of the choices, you could have an edge in the competition. After tallying the previous matches, you nd the counts are as in the following table: Hand Shape Rock Paper Sciss0rs Use the chi-squared goodness of t test to determine if there is evidence that one of the hand shapes is used more frequently than the others. (a) What is H0 and Ha in this case? State it very precisely. (b) What is the chi-squared statistic for the test you carry out, and the corresponding p- value? (c) Write a clear conclusion from the test, and how you would advise a player in the tour- nament based on your ndings. Remember, the player does not understand statistics. (d) (Optional extension of Original problem) Suppose our competitor wins the tournament and hires us to advise him in future tournaments. We use the same approach to tally our ndings for three more tournaments and it yields the results in the following table. Hand Shape Tournament 1 Tournament 2 Rock Paper Scissors Suppose in tournament 1, contestants used their right hand to signal, but in tournament 2, the left hand was used. you want to expand your model to account for which hand was used. Carry out the same procedure from (a), (b), and (c) for this new situation, to nd if right vs. left hand usage has an impact on the proportions of rock, paper, or SClSSOI'S. Solution (a) Ho : Pr = Pp = Ps = 3, H1 : Pi # 5 for some i. Note: if you answered " Ha : Pr # Pp # Ps" this does not capture all of the possibilities, so is incorrect. (b) x 2 E (Oi - Ei)2 Ei where E; is 450/3 = 150, and O; is each of the counts 131, 143, 176 respectively. = 2.41 + .33 + 1.51 = 7.25 The corresponding p-value for a chi-squared statistic of 7.25 with 2 degrees of freedom is .0266, which is significant at the a = .05 level.(c) According to my results, there the odds of scissors are higher than the other hand shapes. Therefore, I recommend you play rock in your next match. (d) (i) We want to test Hg: proportion of hand signs is independent of which hand is used Ha: proportion of hand signs is not independent of which hand is used. (ii) The chi-squared statistic will be calculated in the same fashion where 2 (O, Ea)? X Z T. So Our chi-squared statistic is computed as 6.21. And with degrees of freedom of 4, our p value is .1847, which is not signicant at the a = .05 level. (iii) According to our results, there is not signicant evidence to suggest that the propor- tion of rock, paper, and scissors is not independent of which hand is used. Therefore, we will not choose different play styles in the different tournaments