You play a game of dice at the state fair, but suspect the dice may have been rigged. You observe the game owner rolling the dice 78 times and note the following results:

Face"1""2""3""4""5""6"

Observed count17193121512

(a) Which hypotheses should be used to test if the dice are fair:

Identify H₀:

H₀: p₁= p₂= p₃= p₄= p₅= p₆= 1/6, implying the dice are fair.

H₀: p₁= p₂= p₃= p₄= p₅= p₆= 0, implying the dice are fair.

H₀: p₁= p₂= p₃= p₄= p₅= p₆= 0, implying the dice are unfair.

H₀: p₁= p₂= p₃= p₄= p₅= p₆= 1/6, implying the dice are unfair.

Identify H_a:

H_a: at least one p1/6, implying the dice are unfair.

H_a: at least one p1/6, implying the dice are fair.

H_a: every p1/6, implying the dice are unfair.

H_a: every p1/6, implying the dice are fair.

(b) What is the expected count for each side of the dice?

(c) Is the cell count condition met?

No, because some cells have less than 10 observed counts.

Yes, because the expected cell count is at least 5 for all cells.

Yes, because the expected cell count is close to the observed cell count for all cells.

No, because some cells have less than 30 observed counts.

(d) Calculate the contribution to the test statistic for the dice side with number "1": (Use 4 decimals):

(e) The test statistic is²= 12.154. What is the P-value? (Use 4 decimals.)

(f) What is the appropriate test conclusion, for= 0.05?

There is insufficient evidence the dice are unfair.

There is sufficient evidence the dice are unfair.

There is sufficient evidence the dice are fair.

There is insufficient evidence the dice are fair.