Section 11.2 Hints

Problems 9-17: You will follow the pattern shown below for #9, with extra comments made in red.

#9. The given observed data is as follows (simply copied from the problem):

27 | 31 | 42 | 40 | 28 | 32 |

If the outcomes are equally likely, then the expected data (in 200 tosses) would be: (that is, dividing 200 evenly across 6 cells is 200/6 in each)

33.3 | 33.3 | 33.3 | 33.3 | 33.3 | 33.3 |

Our test:

H_{0}: The data is uniformly distributed

H_{1}: The data is not uniformly distributed

α = 0.05 (this was given)

DR: If χ^{2 }> 11.070, then rej H_{0}. (11.070 comes from the table, using 6-1 degrees of freedom and α = 0.05)

Our statistic is χ^{2 }= Σ(O-E)^{2}/E = 6.3^{2}/33.3 + 2.3^{2}/33.3 + 8.7^{2}/33.3 + 6.7^{2}/33.3 + 5.3^{2}/33.3 + 1.3^{2}/33.3

= 195.34/33.3 = 5.866 (The 6.3 comes from the first cells: 27-33.3 = -6.7, but I left off the minus since it will be squared out. Then the same for the remaining cells.)

Conclusion: Fail to reject H_{0}. The data is uniformly distributed. (The 5.866 is NOT above 11.070, so fail to reject H_{0}.)

Problems 18-20: You will follow the same pattern as above, except you will need to calculate the expected data according to the suggested pattern. Also, the hypotheses will need to be adjusted, as will the 2nd conclusion.

#18. There we 149 data points in the observed data (132+17+0). There was a suggested proportion distribution of 87%, 8% & 5%. So we expect 87% of the 149 data points to be in the first group (that's 149 x 0.87 = 129.63), and 8% in the second group (149 x 0.08 = 11.92), and 5% in the third (149 x 0.05 = 7.45). That's our expected data, which we put into the expected chart, as follows:

129.63 | 11.92 | 7.45 |

Our H_{0} must change to something like "The data is distributed according to the suggested proportions," while
H_{1} could be "The data is not distributed according to the suggested proportions."

And our conclusions will be either "Reject H_{0}. The data is NOT distributed according to the suggested proportions" or "Fail to reject H_{0}. The data is distributed according to the suggested proportions."