Question 1
(a)    Properties of a normal distribution:
1.      If , a and b are real numbers, then ;
2.      If  and  are statistically independent normal random variables, then:
l  The sum of X and Y also satisfy the normal distribution ;
l  The difference of X and Y also satisfy the normal distribution ;
l  U and V are independent of each other.
3.      If  are independent standard normal random variables, then  obey the chi-square distribution which has N degrees of freedom.
(b)   , therefore, . it can be seen that . Then , . According to the Standard normal distribution table, k=-1.96
(c)    (i) , therefore, 1-0.9732=2.68%
(ii)
(iii)
(iv)
Question 2
(a)   
(b)   m=224, n=61,
(i)          95% confidence interval

(ii)        t distribution in 60 degree of freedom,

(iii)      In large-sample case, the t-distribution provides more conservative confidence intervals, which may leads to the results not very accuracy.
Question 3
(a)    According to the question,
Then the 99% confidence interval is

(b)   According to the question,
Therefore, 95% confidence interval is
It can be seen, that the lower limit is larger than 4528, therefore, there ahs been significant increase in sales volume.
Question 4
(a)    Central Limit Theorem
In selecting simple random samples of size n from a population, the sampling distribution of the sample mean  can be approximated by a normal probability distribution as the sample size become large.
(b)  
(i)          point estimate for

(ii)        98% confident maximum error

(iii)      98% confidence interval

(c)    According to the question,

As
Question 5
(a)    Regression analysis and correlation analysis are complementary and closely linked, correlation analysis requires specific forms of regression analysis show the relationship between the phenomena of the numbers, and the regression analysis should base on the correlation analysis. The main differences are: first, in regression analysis, not only depends on the status of variables to distinguish from the role of different variables and the dependent variables, put the dependent variable into the interpreted special status; however, in correlation analysis, the status of variables are completely equal, and all relevant variables are random variables. Second, the correlation analysis only describes the closeness of the relationship of interdependence between variables; however, the regression analysis not only quantitatively reveals the size of the impact about independent variables affecting the corresponding variables, but also predicts and controls the variables via regression equation.
(b)   (i) slope
Intercept


(ii) Coefficient of Determination

Correlation coefficient

According to the above data, it can be seen that there is a reverse relationship between number of houses sold and interest rate.
As SST=SSR+SSE, in this equation, SSE, i.e. sum of squares due to error, is not explained by the regression line.
   (iii.) According to the above analysis, it can be seen that there is a reverse relationship between the sales of houses and interest rate. When the interest rate goes up, the number of house sales decreased; when the interest rate goes down, the number increased. However, there is a special circumstance exist, when the interest rate changed from 11% to 11.9%, the number of house sales does not decrease. Therefore, there may other factors effecting the house sales need to be focused.