Consider a population having a standard deviation equal to 10.05. We wish to est
ID: 3130385 • Letter: C
Question
Consider a population having a standard deviation equal to 10.05. We wish to estimate the mean of this population.
(a) How large a random sample is needed to construct a 95 percent confidence interval for the mean of this population with a margin of error equal to 1? (Round your answer to the next whole number.)
The random sample is__________ units.
(b) Suppose that we now take a random sample of the size we have determined in part a. If we obtain a sample mean equal to 252, calculate the 95 percent confidence interval for the population mean. What is the interval’s margin of error? (Round your answers to the nearest whole number.)
The 95 percent confidence interval is [_____,_____ ] .
Margin of error:_______
Explanation / Answer
a)
Note that
n = z(alpha/2)^2 s^2 / E^2
where
alpha/2 = (1 - confidence level)/2 = 0.025
Using a table/technology,
z(alpha/2) = 1.959963985
Also,
s = sample standard deviation = 10.05
E = margin of error = 1
Thus,
n = 387.9969445
Rounding up,
n = 388 [ANSWER]
******************
b)
Note that
Margin of Error E = z(alpha/2) * s / sqrt(n)
Lower Bound = X - z(alpha/2) * s / sqrt(n)
Upper Bound = X + z(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.025
X = sample mean = 252
z(alpha/2) = critical z for the confidence interval = 1.959963985
s = sample standard deviation = 10.05
n = sample size = 388
Lower bound = 251.0000039
Upper bound = 252.9999961
Thus, the confidence interval is
( 251 , 253 ) [ANSWER, CONFIDENCE INTERVAL]
Thus,
Margin of Error E = 0.999996063 = 1 [ANSWER]
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