Question 1199228: We sampled 800 NY teachers for their annual income. In our sample,
the average annual income is $60,000 and the standard deviation of annual
incomes is $7,500. Based on that sample, we want to predict the mean annual
income μ of all the NY teachers with the confidence level of 0.99 (99 percent).
Part a) What prediction do we make about μ?
Part b) What is the error bound/margin E in that prediction?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sample size is 800
sample mean is 60,000.
sample standard deviation = 7500.
two tailed confidence interval is 99% = .99
alpha on each end of the confidence interval is .5% = .005
for 99% confidence interval, the critical t-score, at 799 degrees of confidence, is plus or minus 2.581997 rounded to 6 decimal places.
the standard error of the distribution of sample means is equal to the standard deviation of the sample divided by the square root of the sample size.
the standard error is therefore 7500/sqrt(800) = 265.165 rounded to 3 decimal places.
use the critical t-score to find the critical sample mean score.
the formula is t = (x - m) / s
t is the t-score
x is the sample mean
m is the assumed population mean
s is the standard error.
you are looking for the margin of error.
that would be represented by (x - m)
on the low side, the t-score formula becomes:
-2.581997 = (60,000 - m) / 265.165
solve for 60,000 - m to get (60,000 - m) = -684.655 rounded to 3 decimal places.
on the high side, the t-score formula becomes:
2.581997 = (60,000 - m) / 265.165
solve for 60,000 - m to get (60,000 - m) = 684.655
your margin of error is 684.655 on both sides of the assumed population mean.
that would make the estimate of the population mean between (60,000 - 685) and (60,000 + 685) = 59315 and 60685.
i don't have a t-score graphing calculator that will show the raw score analysis on a graph, but i do have a z-score calculator that will do that, after manipulation of the confidence interval to make the t-score look like the z-score.
this is what it looks like.
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