全文链接：https://tecdat.cn/?p=33336

Problems:

1. Let X1 and X2 constitute a random sample of size 2 from the population given by
   f(x; θ) = θxθ-1 10;1:
   If the critical region x1x2 ≥ 3=4 is used to test the null hypothesis θ = 1 against the alternative hypothesis θ = 2, what is the power of this test at θ = 2?
2. A random sample of size n is to be used to test the null hypothesis that the parameter
   θ of an exponential population equals θ0 against the alternative that it does not.
   • Find an expression for the likelihood ratio statistic.
   • Use this result to show that the critical region of the likelihood ratio test can be
   written
   x · e-x=θ0 ≤ K:
3. A random sample of size n from a normal population with unknown mean and variance
   is to be used to test the null hypothesis µ = µ0 against the alternative µ 6= µ0. Using
   the simultaneous maximum likelihood estimators of µ and σ2, show that the values of
   the likelihood ratio statistic (LRT) can be written
   λ = 1 + n t-2 1-n=2;
   where t = pn(x - µ0)=s. How is the LRT connected to the t distribution?
4. Independent random samples of sizes n1; n2; · · · ; nk from k normal populations with
   unknown means and variances are to be used to test the null hypothesis that all the
   variances are equal.
   • Derive, under the null hypothesis, the maximum likelihood estimates of the means
   µi and variances σi2. Repeat the derivation with no restriction on the variances (i.e.,
   where the null is not true).
   • Using these results, derive the likelihood ratio test statistic.
   Computer:
5. Write an R script to do the following:
   • Generate a sample of 50 observations from a random N(12; 7) population.
   • Test the null hypothesis that the mean is 3 at a 5 % significance level.
   • Test the null hypothesis that the mean is 11.75 at a 1 % significance level.

• Test the null hypothesis that the variance 9 at a 10 % significance level and at a 1
% significance level.
• Repeat steps 2-4 above for a new sample (generated via the same process) but with
only 10 observations.
• Construct another sample (from the same population) with 50 observations. Test the
hypothesis that both samples of 50 observations each came from normal populations
with the same mean (use α = 5%)

mu = 3
sd = sqrt(7)


x <- rnorm(50)*sd + 12#Generate a sample of 50 observations from a random N (12; 7) population
t.test(x, mu = 3)#Test the null hypothesis that the mean
t.test(x, mu = 11.75,conf.level =0.99 )

var.test(x,rnorm(50,12,3),ratio=1,conf.level=0.9)#Test the null hypothesis that the variance