standard deviation = 7.5
25th percentile means that 25% of the area under the normal distribution curve is to the left of the z-score indicated.
use a z-score calculator to find that the z-score for that is equal to -.6744897...
since z-score = (raw score minus mean) / standard deviation, you get:
-.6744897... = (x - 121.3) / 7.5
solve for x to get x = 116.2423269... which you can round to 116.2.
95th percent confidence interval is calculated as follows:
two sided confidence interval alpha = (100% - 95%) / 200 = .025
the z-scores correspond to that will be a z-score that has 2.5% of the area under the normal distribution curve to the left of it, and a z-score that has 97.5% of the area under the normal distribution curve to the left of it.
to find the confidence interval, you then subtract the smaller area from the bigger area to get the area in between.
the z-score for an area of 2.5% is equal to -1.959963...
the z-score for an area of 97.5% is equal to 1.959963...
the area in between is equal to 97.5 - 2.5 = 905% of the area unde rthe normal distribution curve.
you need to translate the z-scores to raw score.
a z-score of -9.959963... will yield a raw score as follows:
-9.959963... = (x - 121.3) / 7.5
solve for x to get x = 106.6002701...
a z-score of 9.959963... will yield a raw score as follows:
9.959963... = (x - 121.3) / 7.5
solve for x to get x = 135.9997299...
round these to the nearest one decimal place and you get:
95% confidence interval is from 106.6 people to 136.0 people.
you can see these visually from the following pictures.
any differences between the numbers shown in the pictures and the numbers i calculated are due to differences in the way the calculators handle the numbers.
some calculators are consistent with each other, but not all.
differences are usually due to rounding assumptions and to interpolation assumptions.
