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

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.

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

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

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

(6%) (a) the sample mean x̄ = is an unbiased estimator of the population mean μ, and

(9%) (b) the sample variance s2 = is an unbiased estimator of the population variance σ2.

(a) Can the company conclude that Provider A is better than Provider B at a 5% level of significance? (10%)