(iv) (5%) Given x=2, what is the probability that y=10?

1. (10%) Show that the following definition of sample variance s² is a biased estimator of the population variance σ².

(i) (10%) Find the marginal probability mass functions for both X and Y.

(ii) (5%) What is the expected value for Y?

(iii) (10%) Are X and Y statistically independent? Give your reasons.

(i) (10%) What is the probability that there will be exactly one large-magnitude earthquake next year in City A?

(ii) (10%) What is the probability that the next year would be a hazard year for City A?

(iii) (10%) What is the probability that there will be 2 hazard years within the next 4 years?

4. (10%) A survey is performed to study the proportion of residents in City A and City B who support the construction of a new railway line. It is found that 680 of 1000 City A residents favor the construction while 730 of 1000 City B residents are in favor. Is there a significant difference between the proportions of City A and City B residents who favor the construction of the new railway line? Use the P-value approach to help your draw your conclusion. Use 0.05 as the level of significance.

(i) Construct a 95% confidence interval for the difference in the mean yield strength for the 2 companies. Assume normality for the distribution of yield strength. (10%)

(ii) Construct a 95% confidence interval for the population variance of the yield strength for Company A. (10%)