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.
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
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