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QUESTION 06 (10 points) - Goodness-of-Fit Test of the Poisson Distribution You will need to test if the following sample comes from a Poisson distribution at ? = 0.07. Category x P(x) Observed Frequency (no observations for x = 7 or above) 1 0 ? 41 2 1 ? 115 3 2 ? 119 4 3 ? 103 5 4 0.12258 49 6 5 ? 22 7 6+ ? 26 TOTAL 1.00000 475 (a) [1] Form the null and alternative hypotheses H0 and H1. Make a sketch of the test. (b) [1] Specify appropriate equations for the ?² statistic and degrees of freedom df. (c) [1] Estimate ? as the mean of a Poisson distribution if H0 holds true. Thus ?? = x?. (see Hint) (d) [1] Given ??, use Excel to calculate expected probabilities for x values. (see Hint) (e) [1] Based on the expected probabilities, calculate expected frequencies. (f) [1] Based on expected and observed frequencies, calculate the ?² statistic and its df. (g) [1] Given ? = 0.07 and df, use CHISQ.INV(?,df) to look up the critical value ?². (h) [2] Compare ?² statistic with ?² critical to accept or reject H0. Interpret your result. (i) [1] Sketch and find the p-value of the test. Would you reject the null if ? = 0.03? Hint: ?? = 2.36632 (check why). Use POISSON.DIST(x,??,0) to calculate expected probabilities. Some probabilities are pre-calculated for your reference. Refer to some Excel lookups: ?v 0.96 0.93 0.93 0.93 0.90 ?² 10.46170 9.41553 11.50787 12.55404 10.46170 df 5 7 6 5 6 CHISQ.INV(?v,df) 11.64433 13.08771 11.65992 10.19103 10.64464 CHISQ.DIST(?²,df,1) 0.93684 0.77582 0.92611 0.97206 0.89349

          QUESTION 06 (10 points) - Goodness-of-Fit Test of the Poisson Distribution
You will need to test if the following sample comes from a Poisson distribution at ? = 0.07.

Category     x     P(x)     Observed Frequency (no observations for x = 7 or above)
1            0     ?        41
2            1     ?        115
3            2     ?        119
4            3     ?        103
5            4     0.12258  49
6            5     ?        22
7            6+    ?        26

TOTAL              1.00000  475

(a) [1] Form the null and alternative hypotheses H0 and H1. Make a sketch of the test.
(b) [1] Specify appropriate equations for the ?² statistic and degrees of freedom df.
(c) [1] Estimate ? as the mean of a Poisson distribution if H0 holds true. Thus ?? = x?. (see Hint)
(d) [1] Given ??, use Excel to calculate expected probabilities for x values. (see Hint)
(e) [1] Based on the expected probabilities, calculate expected frequencies.
(f) [1] Based on expected and observed frequencies, calculate the ?² statistic and its df.
(g) [1] Given ? = 0.07 and df, use CHISQ.INV(?,df) to look up the critical value ?²*.
(h) [2] Compare ?² statistic with ?²* critical to accept or reject H0. Interpret your result.
(i) [1] Sketch and find the p-value of the test. Would you reject the null if ? = 0.03?

Hint: ?? = 2.36632 (check why). Use POISSON.DIST(x,??,0) to calculate expected probabilities.
Some probabilities are pre-calculated for your reference. Refer to some Excel lookups:
?v    0.96   0.93   0.93   0.93   0.90
?²    10.46170 9.41553 11.50787 12.55404 10.46170
df    5       7       6       5       6

CHISQ.INV(?v,df)   11.64433 13.08771 11.65992 10.19103 10.64464
CHISQ.DIST(?²,df,1) 0.93684  0.77582  0.92611  0.97206  0.89349

QUESTION 06 (10 points) - Goodness-of-Fit Test of the Poisson Distribution
You will need to test if the following sample comes from a Poisson distribution at ? = 0.07.

Category x P(x) Observed Frequency (no observations for x = 7 or above)
1 0 ? 41
2 1 ? 115
3 2 ? 119
4 3 ? 103
5 4 0.12258 49
6 5 ? 22
7 6+ ? 26

TOTAL 1.00000 475

(a) [1] Form the null and alternative hypotheses H0 and H1. Make a sketch of the test.
(b) [1] Specify appropriate equations for the ?² statistic and degrees of freedom df.
(c) [1] Estimate ? as the mean of a Poisson distribution if H0 holds true. Thus ?? = x?. (see Hint)
(d) [1] Given ??, use Excel to calculate expected probabilities for x values. (see Hint)
(e) [1] Based on the expected probabilities, calculate expected frequencies.
(f) [1] Based on expected and observed frequencies, calculate the ?² statistic and its df.
(g) [1] Given ? = 0.07 and df, use CHISQ.INV(?,df) to look up the critical value ?²*.
(h) [2] Compare ?² statistic with ?²* critical to accept or reject H0. Interpret your result.
(i) [1] Sketch and find the p-value of the test. Would you reject the null if ? = 0.03?

Hint: ?? = 2.36632 (check why). Use POISSON.DIST(x,??,0) to calculate expected probabilities.
Some probabilities are pre-calculated for your reference. Refer to some Excel lookups:
?v 0.96 0.93 0.93 0.93 0.90
?² 10.46170 9.41553 11.50787 12.55404 10.46170
df 5 7 6 5 6

CHISQ.INV(?v,df) 11.64433 13.08771 11.65992 10.19103 10.64464
CHISQ.DIST(?²,df,1) 0.93684 0.77582 0.92611 0.97206 0.89349

Added by Irene T.

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