(a) What criteria are used to select a better estimator? (3%)

5. (12%) Suppose that we have independent random samples of n₁, n₂,..., nₖ observations from K population. Denote by..., the K group sample means and by the overall sample mean. We define the following sums of squares:Within group sum of squares (SSW) =Between groups sum of squares (SSG) =Total sum of squares (SST) = where Xᵢⱼ denotes the jᵗʰ sample observation in the iᵗʰ group. Please show the followingrelationshipSST = SSW + SSG

(6-A)Test whether opinion and gender of viewer are independent, controlling the α risk at 0.10. State the alternatives, the decision rule, the value of test statistic, and the conclusion. (12%)

(6-B) Given the conclusion is part (6-A), should the next step be to examine the relative squared residuals for the test? Explain. (4%)

1. (10%) Please present the figures with symmetric, right-skewed and left-skewed distribution respectively, and indicate the relative locations of mean, median and mode under such three types of distribution.

(b) What is the Type I (α Type) error? (3%)

(c) What parameters are required to specify an F-distribution function? (3%)

(d) What are the expectation value and variance for the random variable following the Chi-square () distribution with k degrees of freedom? (3%)

(e) What is the simple random sampling? (3%)

3. (10%) When the production process is functioning correctly, which is 90% of the time, 10% of all bulbs produced are defective. However, the process is susceptible to an occasional malfunction, leading to a defective rate of 50%. If a defective bulb is found, what is the probability that the process is functioning correctly? If a non-defective bulb is found, what is the probability that the process is operating correctly?

4. (10%) A random sample of 100 people shows that 25 are left-handed. Please form a 95% confidence interval for the true proportion of left-handers