At 95% confidence, the z critical value is roughly z = 1.96
You would use a reference table or calculator to determine this value. Unfortunately it's not something that can be easily/quickly found by hand.
If we're trying to estimate the population mean (mu), then we'll use this margin of error formula
E = z*sigma/sqrt(n)
Or if we want to estimate the population proportion (p), then we use this formula
E = z*sqrt(phat*(1-phat)/n)
The confidence interval then spans from (pointEstimate)-E to (pointEstimate)+E. Where "pointEstimate" represents either xbar or phat depending on the context. Other point estimates are possible. We're focusing on these two for now.
xbar estimates mu
phat estimates p
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An example problem:
A manager at a soda bottling plant wants to know if the contents of the cans are 20 fluid ounces (fl oz) or not. Through past observations, the manager has noted the population standard deviation (sigma) is 1.72 fl oz.
The manager gathers a random sample of 50 cans to test their contents. They find the sample mean is 18 fl oz. Conduct a hypothesis test using a confidence interval at 95% confidence to see if the cans really do have 20 fl oz of drink in them or not.
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Solution:
Null Hypothesis: mu = 20
Alternate Hypothesis: mu ≠ 20
This is a two-tailed test.
E = margin of error
E = z*sigma/sqrt(n)
E = 1.96*1.72/sqrt(50)
E = 0.47676
L = lower boundary of confidence interval
L = xbar - E
L = 18 - 0.47676
L = 17.52324
L = 17.52
U = upper boundary of confidence interval
U = xbar + E
U = 18 + 0.47676
U = 18.47676
U = 18.48
The 95% confidence interval is approximately (17.52, 18.48)
This is equivalent to writing 17.52 < mu < 18.48 in which we are 95% confidence in saying mu is between those endpoints.
But wait, the hypothesized mu = 20 is not in the interval 17.52 < mu < 18.48, which indicates we must reject the null hypothesis and accept the alternate hypothesis.
The soda cans do not have 20 fl oz of drink. It appears they have slightly less than 20 fl oz.