I can calculate part a but I don't really understand part b. Is part b just same formula as part a but change the value of percentiles? How to calculate it? And if comparing part b to part a, Why we commonly use Symmetry setting ($_{/2} _{1−/2}$) in confidence interval?
Also, for part d, as the question asks for minimum sample size, I compute like the following: $\frac{z_{0.95}}{\sqrt n}\times 2$<2, n>$\frac{(2z_{0.95})^2}{2^2}$, then do I just substitute =0.28 and complete the calculation and finally it's the correct answer? Or does equal to other value?
Question
Consider a new proposed antibiotic is taken by 16 patients, and their
serum-creatinine level are measured 24 hours after. Assume that the serum-creatinine level
is normally distributed, and the sample mean of serum-creatinine level was 1.50 mg/dL and
sample variance of serum-creatinine level was 0.0784.
Suppose that the standard deviation of serum-creatinine level in population is known at
0.28 mg/dl.
(a) Construct a 95% Confidence Interval for the mean serum-creatinine level μ.
(b) Instead of using the $z_{0.025} = -1.96 and z_{0.975} =1.96 $to construct the 95%
Confidence Interval for μ in part (a), construct another 95% Confidence Interval for μ
based on $z_{0.02} and z_{0.97}$ (i.e. 2nd and 97th percentile of N (0,1)).
(c) Compare part (b) to part (a), what’s the difference between these two
confidence intervals? Which confidence intervals provide a more accurate estimation?
Why? [Why we commonly use Symmetry setting ($_{/2} _{1−/2}$) in confidence
interval?]
(d) If we want to control the width of 90% confidence interval for is less than 2
(Margin of Error is less than 1), what’s the minimum value of the sample size we
should collect?
Instead of $z = -1.96$ and $z = 1.96$ as the critical values, which correspond to a lower tail area of $0.025$ and upper tail area of $0.025$, they are asking for $z_{0.02}$, which would correspond to the left edge of a $96$% confidence interval, and $z_{0.97}$, which would be the right edge of a $94$% confidence interval.
I don't know what tool you use to look up those values. There are lots of software options, calculators, and tables.
On a TI-83+, for example, you can use DISTR to get the invnorm function and enter .02 and then do it again with 0.97, and it should give you $z$ values. In particular, I get $-2.0537$ and $1.8808$ respectively.
You can also find those values in a $z$-table, by searching in the center mass of numbers for $0.02$ (or $0.98$ if your table only uses positive values of $z$) and $0.97$.