3. Let X,Xn iid Unif(o,0) . Is this family MLR in Y = Xo)?(b) Find the UMP size-a test for Ho : 0 < 0 vs H, : 0 > 0 Find the UMP size-a test for Ho : 0 2 0o v...

(a) identify the expected distribution and state $H_{0}$ and $H_{a}$, (b) find the critical value and identify the rejection region, $(c)$ find the chi-square test statistic, $(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. An organization claims that the number of prospective home buyers who want their next house to be larger, smaller, or the same size as their current house is not uniformly distributed. To test this claim, you randomly select 800 prospective home buyers and ask them what size they want their next house to be. The table at the left shows the results. At $\alpha=0.05,$ test the organization's claim.

Here we are asked to find the mean and variance of the distribution defined by this probability density function here. Now we can see that this is the pdf of a gamma of a gand um gamma random variable. And I provided the general pdf for the gamma random variable here, so we can see by comparing this to the general form that are is equal to three, and we can also see that lambda Is equal 2.01 for a gamma random variable, the mean is equal to our overland, to Which in this case comes out to 300 and the variance is equal to our overland squared, And this comes out to 30,000. So we have a mean of 300 And the variance of 30,000.