How To Use SATA In solving Stat Problem?

Problem1:

Let X_i (i=1,2,…,n) be a random sample for Poisson(?₁) and Y_j (j=1,2,…,m) be a random sample for Poisson(?₂). Assume that the sample are independent.

1. Consider H₀: ?₁= ?₂ vs H₁: ?₁? ?₂

Assuming that the sample sizes are large, derive the simplest form of rejection region for large sample LRT of size a.

1. The following samples are from two Poisson distributions. Use part (a) to determine whether or not the mean of the Poisson distributions are equal. Use a=.05.

X

1 0 4 2 0 3 3 0 1 1 0 1 1 1 1 2 5 1 1 0 1 0 0 0 2 1 0 4 1 4 0 2 1 2 3 1 5 2 2 2 2 5 2 1 1 0 2 1 3 1 1 1 2 1 0 1 2 2 3 2

Y

3 1 5 2 1 0 1 2 1 3 2 0 0 4 3 2 0 0 0 5 1 2 0 0 2 3 0 1 1 2 3 3 1 0 1 3 3 2 1 0 3 1 2 1 0 1 0 2 1 0

Problem 2:

Let X_i (i=1,2,…,n) be a random sample for gamma(a,b) and assume that a is known.

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1. Derive the Wald test for H₀: b=b₀ vs H₁: b?b₀. Write the test stat in its simplest form.
2. Derive the Score test for H₀: b=b₀ vs H₁: b?b₀. Write the test stat in its simplest form.
3. A random sample of size 100 from gamma(a=2,b) yields the sample mean 9.479 and the sample standard deviation of 6.812. Test H₀:b=4.8 vs H₁:b?4.8 using the score test at the .05 level of significance.
4. The sorted observations of the sample are below. Using the multinomial distribution and large sample results verify whether or not the sample was generated from gamma(a=2,b). Use .05 level of significance. You may need to use a software that generates gamma cdf.

0.6913 0.9922 1.0955 1.1467 1.2455 1.5184 1.7168

1.8696 2.0051 2.4345 2.5854 2.7481 2.9646 3.2507

3.4182 3.5738 3.5848 3.8285 3.8755 3.8873 4.0553

4.1962 4.2059 4.2743 4.6763 4.8611 4.9076 5.0102

5.0638 5.2174 5.2873 5.3242 5.4229 5.6941 5.8324

5.9102 5.9484 6.0126 6.3358 6.6777 7.1031 7.1974

7.2406 7.5163 7.7496 7.9024 8.1623 8.2032 8.2731

8.3786 8.6828 8.7837 9.3558 9.3838 9.5398 9.6141

9.7111 9.7504 9.8943 9.9800 10.0142 10.2260 10.4963

10.5273 10.9390 11.2684 11.3069 11.3893 11.4734 11.4987

11.5170 11.6297 12.0685 12.3209 12.3284 13.0722 13.1213

13.2434 13.6201 13.6610 13.9758 14.0619 14.1057 14.1605

14.1615 14.1751 14.4600 14.6078 15.9495 16.0973 16.9015

16.9538 17.8670 17.8736 19.8112 20.1432 28.0960 29.1380

31.1903 42.6680