Part A:

For any confidence interval, call value of the confidence level (such as 90% or 95%, as in your case). Then the probability that the true population mean (called ) lies within a given interval with a confidence level of is:

where are your mean, z-score, standard deviation, and sample size, respectively.

So your confidence interval is between the points

To find a given confidence interval, plug in those values. In your case, [tex]\bar{x} = 2.9, n = 160, \sigma = 0.1[tex]. The z-score will be based on your confidence level. Look them up in a z-table. A 90% confidence interval will exclude 5% of the normal distribution on neither side (a total of 10% excluded). A 95% confidence interval will exclude 2.5% (for a total of 5% excluded).

Part B:

Consider what a confidence interval means. A 90% confidence interval means that repeated trials using those values of will give you intervals that contain the true mean 90% of the time. A 95% confidence will do the same thing 95% of the time. But for a 95% confidence interval, the z-value is higher, so what happens to the width of the interval?

The answer is that both have their pros and cons - you are essentially trading off how certain you are about where the mean is with precise of a net you are casting. The higher the confidence level, the wider the interval is. It's sort of like: would you prefer "I am 95% that this guy is in America" vs. "I am 80% sure this guy is in New York?"