The national 'marbles playing' championships are nearing its conclusion, with only two teams remaining that will face off in the final round. It's a true David and Goliath situation with Team 1 consisting of 8 pre-schoolers, and Team 2 consisting of 6 adult professional marble players. Each team can use the average number of marbles each person won in the previous rounds in the final match and betting offices usually favor the team with the most marbles at the start of the match. Let's figure out if one team starts with significantly more marbles than the other team.

The number of marbles per team member are as follows

X = Team 1: [24, 21, 18, 27, 31, 29, 32, 29]

Y = Team 2: [31, 33, 27, 32, 32, 30]

. Please show how to conduct a two-tailed hypothesis test at alpha=0.05. please provide the following information and show the math

a. State the H₀ and H₁

b. Calculate the difference between sample means

c. Calculate the corrected variance for each sample (S²_x and S²_y)

d. Calculate the total degrees of freedom (df_total)