Question 1192326: it is known that 95% confidence limits to population mean are 48.04 and 51.96 what is the value of population variance when sample is 100? Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! two tailed confidence interval of .95 has critical z-score of plus or minus 1.96.
the 95% confidence limits are 48.04 and 51.96.
the z-score formula is z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard error.
the standard error is equal to standard deviation divided by square root of sample size.
formula is s = sd / sqrt(ss)
s is the standard error
sd is the standard deviation
ss is the sample size
when the sample size is 10, the standard error formula becomes:
s = sd / 10.
the mean is halfway between the low score and the high score.
the low score is 48.04 and the high score is 51.96.
the mean is therefore equal to (51.96 minus 48.04) / 2 = 50.
on the low side of the confidence interval, the z-score formula of z = (x - m) / s becomes:
-1.96 = (48.04 - 50) / (sd / 10)
simplify to get:
-1.96 = -1.96 / (sd / 10)
multiply both sides of this formula by (sd / 10) to get:
-1.96 * sd / 10 = -1.96
divide both sides of this formula by -1.96 and multiply both sides of this 10 to get:
sd = 10
since the variance is equal to the square of the standard deviation, you get variance = 100.
on the high side of the confidence interval, the z-score formula of z = (x - m) / s becomes:
1.96 = (51.96 - 50) / (sd / 10)
simplify to get:
1.96 = 1.96 / (sd / 10)
multiply both sides of this formula by (sd / 10) to get:
1.96 * sd / 10 = 1.96
divide both sides of this formula by 1.96 and multiply both sides of this 10 to get:
sd = 10
since the variance is equal to the square of the standard deviation, you get variance = 100.
your solution appears to be that the population variance is 100 when the sample size is 100 and the lower and upper critical scores are 48.04 and 51.96.
when the standard deviation is 10, then the standard error becomes 10 / sqrt(100) = 10 / 0 = 1.
there's a nice little normal distribution z-score calculator online that will allow you to visualize all this.
here are the displays from that calculator.
the first two are finding the area from the z-score and the raw score.
the second two are finding the z-score and the raw score from the area.
