Suppose there are 3 species of fish in the world, we are interested in two fish

Question

Suppose there are 3 species of fish in the world, we are interested in two fish populations: fish of the pacific ocean and fish of the atlantic ocean. Each one of these populations is characterized by a distribution over the species of fish: The pacific distribution is uniform over the 3 species: Pp=(1/3,1/3,1/3) while in the Atlantic population the first species is twice more common than the other two: Pa=(1/2,1/4,1/4)

Suppose we are given a batch of 65 fish whose source is unknown. The number of fish of each species is 30,20,15.

We would like to know whether the the batch of fish is from the Atlantic or the pacific (or neither)

One way of formalizing this problem is to consider two alternative hypotheses:

HA = the batch is from the Atlantic ocean. More technically, the batch was sampled from the distribution Pa

HP = the batch is from the pacific ocean. More technically, the batch was sampled from the distribution Pb.

The test that we will use is the 2 test.

The formula of the 2-statistic is

2= Npi * ((Oi/N pi) / pi)2 (from i=1 to n)

Where: n is the number of species (3 in our case). pi is the probability of the ith species according to the Null distribution. Oi is the number of observations of species i N is the total number of observation. N=Oi (from i = 1 to n)

Given N = 65,

Write expressions for the 2 statistic for the two null hypotheses:

2) For hypothesis HA ( batch is from Atlantic ocean), 2= (decimal number with 6+ decimal places)

3) For hypothesis HP ( batch is from pacific ocean), 2= (decimal number with 6+ decimal places)

Explanation / Answer

here level of singificance is not given . let it is alpha=0.1

null hypothesis H0:observe frequency=expeced frequency

alternate hypothesis H1: observe frequency is not =expeced frequency

there are 3 class so df for chi-square 2 and critical value chi-squar(0.1,2)=4.61

since calcuated chi-squar=5.38 is more than critical chi-square so we fail to accetp H0 and conclude that population is not from pacicific

since calcuated chi-squar=1.15 is less than critical chi-square so we acceot H0 and conclude that population is from atlantic

for pacific observed (O) Expected(E) (O-E) (O-E)2/E 30 21.66666667 8.333333 3.205128 20 21.66666667 -1.66667 0.128205 15 21.66666667 -6.66667 2.051282 65 65 0 5.384615

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