help! intervals and bell curves

Question

A distribution of values is normal with a mean of 200 and a standard deviation of 19. From this distribution, you are drawing samples of size 22.

Find the interval containing the middle-most 32% of sample means: 

Enter your answer using interval notation in the form (a,b). In this context, either inclusive or exclusive intervals would be acceptable. Your numbers should be accurate to 1 decimal places. Answers obtained using z-scores rounded to 2 decimal places are accepted.

Confused on the finding of z score and what to do when drawing a normal curve.

Answer 1

The middle 32% of the distribution of sample means will be from the 50 +/- 16 percentiles, or from 34th to 66th percentiles.

The z-score for the 66th percentile is 0.41.

xbar = sample mean

pop men = 200

std dev = 19

sample size = 22

standard error (SE) = 19/sqrt(22) = 4.05

find xbar corresponding to 66th percentile (z = 0.41):

z = (xbar - pop mean)/SE

0.41 = (xbar - 200)/4.05

xbar = 4.05 * 0.41 + 200 = 201.66

that is the upper bound of the interval

it is 201.66 - 200 = 1.66 above the population mean

since the normal distribution is symmetric, the lower bound of the interval will be 1.66 below the population mean = 200 - 1.66 = 198.34

So the interval is (198.34, 201.66)