Statistics Confidence Intervals - Don't Understand Solutions So I have solutions to two statistics questions below, but I don't quite understand where some values came from and was hoping someone could clarify. I bolded the steps I didn't understand and also left a comment at the start of each solution. 
Q1:
An estimate of the percentage of the defectives in a lot of pins supplied by a vendor is desired to be within 1% of the true proportion at 90% confidence level. 
(b) If the actual percentage of the defectives is unknown, what is the minimum sample size needed for the study?
The solution is below but 
I don't understand why p = 0.5? What makes that the worst case?
so worst case for the value of sqrt( p*(1-p)/n) is when p =0.5 
CI_Low = p - Z_critical* sqrt( (p)* (1-p)/ n) 
CI_High = p + Z_Critical sqrt( p(1-p)/n) 
1.645*sqrt ( p*(1-p)/ n) = 0.01 
sqrt( 0.5*0.5/n) = 0.01/1.645 = 0.006079027 
0.25/n = 3.69546E-05 
3.69546E-05*n = 0.25 
n = 0.25/3.69546E-05 = 6765.0625 
n = 6766 
Q2: 
A statistician estimates the 92% confidence interval for the mean of a normally distributed population as (162.75, 173.25) at the end of a sampling experiment assuming a known population standard deviation. 
a. Use the information given to construct the 97% confidence interval for the population mean. 
The solution for this is long so I'm not going to paste all of it, but I was wondering why the tails '4%' and '1.5%' need to be added to the critical z values? I tried searching online but I couldn't figure out what formula or rule this falls under?
CI_Low = mean - Z_critical*standard deviation/sqrt(N) 
CI_High = mean + Z_critical*standard deviationa/sqrt(N) 
mean = (162.75 +173.25) /2 = 168  
92 % confidence range has 4 % tail on both sides 
Z_critical =  0.96
Z_critical = 1.750686071 
P(z< 1.75 ) = 0.9599 
P(z< 1.76) = 0.9608 
so for 97 % confidence range 
97 % has 1.5 % tails on both sides 
P(z< Z) = 0.985 gives Z_critical for that 
P(z<2.17) = 0.9850 
 A: If you want a better, quicker answer, try asking on http://stats.stackexchange.com
On Q1, p=.5 is the worst case scenario in terms of variance, since, with $n$ fixed, a binomial distribution $B(n,p)$ will reach maximum standard deviation for p=1/2
The formula for the second question is just the way to calculate the shortest interval that contains 97% of all observations from a normal distribution,or, if you prefer, an interval with its center on the mean that contains 97% of all observations.
Ssample mean always approaches a normal distribution for large enough samples, according to the Central Limit Theorem, with the mean being the actual population mean and the standard deviation converging towards $\sigma / \sqrt n$ where $\sigma$ represents the standard deviation for a single observation.
A: Thoughtful question. I will try to answer just the parts you put in bold type.
(1) Here is a plot of $p(1-p)$ against $p$ for $0 < p < 1.$

Notice that the maximum of this parabolic curve is at $p = 1/2.$ Consequently, the standard
error $\sqrt{p(1-p)/n}$ is maximized when $p = 1/2.$ 
Because
you don't know $p,$ you need to use the largest possible standard
error in order to be sure the sample size $n$ is large enough.
(2) A 92% confidence interval is of the form $\bar X \pm 1.7507\frac{\sigma}{\sqrt{n}}.$ The bold statement has to do with
how the numbers $\pm 1.7507$ are found. In order to have 92% of the probability under the standard normal curve below (in the central part), you need to put 4% of the probability in each tail--that is outside of the vertical dotted lines at each end of the curve.[Similarly, for a 95% CI, you put probability 2.5% in each tail; for a 90% CI you put 5% in each tail; and for a 97% CI you put 1.5% in each tail.]

