a) Suppose we observe the following random sample of size n=117, sampled from a discrete distribution. Value 0 1 2 3 4 5 6 Count 32 36 21 18 6 2 2 We would like to test whether this data comes from a Poisson(λ) distribution. The first step is to estimate the mean λ. Compute an estimate of λ using the provided data. Give answer to three decimal places. b)We next estimate the expected counts for each observed data value, assuming the data comes from a Poisson distribution with mean λ, where λ is the mean you just computed. Denote by Ek with k=0,…6 the expected number of k's you expect to observe. What is E0? Give answer to three decimal places. c) What is E1? Give answer to three decimal places. d)What is E2? Give answer to three decimal places. e)What is E3? Give answer to three decimal places. f)What is E4? Give answer to three decimal places. g) What is E5? Give answer to three decimal places. h) What is E6? Give answer to three decimal places. i) Next, take the values of Ek you just computed, and calculate the observed value of the appropriate Goodness of Fit test statistic, used to test the null hypothesis that the observed data is Poisson distributed. Give answer to three decimal places. j) What is the distribution of this test statistic, assuming H0 is true? k)Compute the p-value of the observed test statistic you just computed. Give answer to three decimal places. l)Consider testing ?0H0 at the 5% significance level. Do you reject the null hypothesis?
a) Suppose we observe the following random sample of size n=117, sampled from a discrete distribution.
| Value | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Count | 32 | 36 | 21 | 18 | 6 | 2 | 2 |
We would like to test whether this data comes from a Poisson(λ) distribution.
The first step is to estimate the mean λ. Compute an estimate of λ using the provided data. Give answer to three decimal places.
b)We next estimate the expected counts for each observed data value, assuming the data comes from a Poisson distribution with mean λ, where λ is the mean you just computed.
Denote by Ek with k=0,…6 the expected number of k's you expect to observe.
What is E0? Give answer to three decimal places.
c) What is E1? Give answer to three decimal places.
d)What is E2? Give answer to three decimal places.
e)What is E3? Give answer to three decimal places.
f)What is E4? Give answer to three decimal places.
g) What is E5? Give answer to three decimal places.
h) What is E6? Give answer to three decimal places.
i) Next, take the values of Ek you just computed, and calculate the observed value of the appropriate Goodness of Fit test statistic, used to test the null hypothesis that the observed data is Poisson distributed.
Give answer to three decimal places.
j) What is the distribution of this test statistic, assuming H0 is true?
k)Compute the p-value of the observed test statistic you just computed. Give answer to three decimal places.
l)Consider testing ?0H0 at the 5% significance level. Do you reject the null hypothesis?
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