After getting trounced by your little brother in a​ children's game, you suspect the die he gave you to roll may be unfair. To​check, you roll it 48 ​times, recording the number of times each face appears. Do these results cast doubt on the​ die's fairness?

Face Count

1 9

2 11

3 6

4 6

5 9

6 7

​a) If the die is​ fair, how many times would you expect each face to​ show?

​b) To see if these results are​ unusual, will you test​goodness-of-fit, homogeneity, or​ independence?

​c) State your hypotheses. Choose the correct answer below.

​d) Check the conditions. Is the counted data condition​met?

​e) How many degrees of freedom are​ there?

​f) Find chiχsquared2 and the​ P-value.