I need help on problem 26.  Please show all work. see sample for homework layout.


The main topic for this chapter is that “xbar (as well as pbar) will approximately follow a normal distribution when sample size n gets larger (usually it means n>=30)”.  What if n is small (i.e., n<30)? In this situation, when will need to know an additional condition to have xbar follow a normal distribution. That is, “when sampling from a normal population or in other words, we have a normal population, then xbar follows a normal distribution for any n (i.e., n is not necesarry to be large as stated above)”. As you can see, pbar will not have this luck (why? since the population X=0 (no), and 1 (yes), does not follow a normal distribution). Therefore, for p-bar to follow a normal distribution, we always have n>=30. In fact, pbar is just another xbar with observations 0, and 1. For instance, pbar is the percentage of students living in Sugar Land from 100 students enrolled at a university. So, every student can be coded as 0 (no) or 1 (yes), and pbar, the % of students living in SL, is the sample mean, xbar, with observations of 0 or 1.

By knowing Xbar (and Pbar) followed a normal distribution, you need to know further that the mean of xbar equals to the population mean, mu, and the s.d. of xbar equals to the population s.d. sigma/sqrt(n), where the population mean and the population s.d. sigma will be given in a question). In other words, for xbar, E(Xbar)=the population mean, mu, and s.d.(Xbar)=sigma/sqrt(n). And, the E(Pbar)=the population mean, p, and the s.d.(Pbar)=sqrt(p*(1-p)/n), since for this special case, the population s.d. sigma=sqrt(p*(1-p)).

After knowing the above theoretic reulsts, the stats functions to find the P(Xbar<a), P(Xbar>b), or P(a<Xbar<b) will be norm.dist providing with proper mean and the s.d. (what are the proper mean and s.d. for xbar? Again, the E(xbar) is in Equation 7.1and the s.d.(xbar) is in Equation 7.2).  Please do not use norm.s.dist in this chapter; otherwise, you are partially manually finding the z, then plug it into the stats function norm.s.dist.