Statistics (Confidence Interval)?
A confidence interval for the average tru stray load loss for a particular type of induction, when the current line to 10 amps at a speed of 1500 rpm desired place. Assuming that the stray-load is normally distributed with σ = 3. 0.A. Calculate a 95% confidence interval for μ when n = 25 = 58 and X-Bar. 3B. Compute a 99% confidence interval for μ when n = 100 = 58 and X-Bar. 3C. How large must n, if the width of its 99% confidence interval for μ to 1. 0?
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Confidence intervals are used to find a region in which we are 100 * ( 1 – α )% confident the true value of the parameter is in the interval.
For large sample confidence intervals about the mean you have:
xBar ± z * sx / sqrt(n)
where xBar is the sample mean
z is the zscore for having α% of the data in the tails, i.e., P( |Z| > z) = α
sx is the sample standard deviation
n is the sample size
The sample mean xbar = 58.3
The sample standard deviation sx = 3
The sample size n = 25
The z score for a 0.95 confidence interval is the z score such that 0.025 is in each tail.
z = 1.959964
The confidence interval is:
( xbar – z * sx / sqrt( n ) , xbar + z * sx / sqrt( n ) )
( 57.12402 , 59.47598 )
== — == — == — == — == — ==
B)
The sample mean xbar = 58.3
The sample standard deviation sx = 3
The sample size n = 100
The z score for a 0.99 confidence interval is the z score such that 0.005 is in each tail.
z = 2.575829
The confidence interval is:
( xbar – z * sx / sqrt( n ) , xbar + z * sx / sqrt( n ) )
( 57.52725 , 59.07275 )
== — == — == — ==
C)
To find the sample size needed for a confidence interval of a given size we need only to concern ourselves with the error term of the CI.
We know that the interval is centered at xbar so we need to find the value of n such that
z * sx / sqrt(n) = width.
The z-score for a 0.99 confidence interval is the value of z such that 0.005 is in each tail of the distribution.
z= 2.575829
The equation we need to solve is: z * sx / sqrt(n) = width
n = (z * sx / width) ^ 2.
n = ( 2.575829 * 3 / 1 ) ^ 2
n = 59.71407
Since n must be integer valued we need to take the ceiling of this solution. Always take the ceiling so that the size of the CI will be correct.
n = 60