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Z-test and Confidence Intervals
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Z-tests and confidence intervals for a population mean using technology, including Z-Test for one sample, two-sample and paired samples via differences with μ0=0, normal confidence intervals when σ is known, t-intervals when σ is unknown, and the use of critical values, critical regions and confidence levels.
Want a deeper conceptual understanding? Try our interactive lesson!
Understanding confidence intervals
AHL AI 4.16
A confidence interval is a range of values in which we believe the true mean of the population lies.
When we take samples from a larger population, our samples tend to cluster around the mean of the population. Using a sample, we can make an educated guess that the true population mean is in some range. The more data we have, the better our guess gets.
We calculate this interval in line with the level of certainty we want to have that our range contains the true mean. Obviously, we can be 100% sure that the value is between −∞ and +∞, but the smaller the range, the lower the confidence level we can have.
Normal confidence interval using technology
AHL AI 4.16
If we know the standard deviation of the population whose mean μ we want a confidence interval for, we use a so called normal confidence interval.
Your calculator should include a statistical test called Zinterval or similar. To use it:
Enter the value of σ, which must be known for a Z-test of any kind.
Enter either
Data: a list of values you've typed into the calculator
Stats: the sample mean xˉ and n, the number of samples.
Enter the confidence level and hit calculate
The calculator returns the desired interval, which is symmetrical around xˉ.
T-interval confidence interval using technology
AHL AI 4.16
To find a confidence interval for the mean without knowing the variance, we first have to use our sample to first estimate the standard deviation. Because of this extra uncertainty, we switch to using a t-distribution.
Your calculator should include a statistical test called Tinterval or similar. To use it:
Enter either
Data: a list of values you've typed into the calculator
Stats: the sample mean xˉ, the sample standard deviation Sx and n, the number of samples.
Enter the confidence level and hit calculate
The calculator returns the desired interval, which is symmetrical around xˉ.
Z-Test for population mean
AHL AI 4.18
Z-tests allow us to test the mean of a sample against
a population with known mean: use Z-Test
another sample: use 2-SampZTest
a paired sample: calculate the difference, then use Z-Test with μ0=0.
When testing for the mean, or finding confidence intervals, we have to decide between using a T-test or a Z-test:
We use Z (normal) when the standard deviation is known.
We use T when the standard deviation is not known.
The T-distribution is very close to the normal, except that is slightly wider to account for the fact that we are estimating the standard deviation using a sample.
Critical values & regions
AHL AI 4.18
When testing the mean of a sample against a population, the critical region is the set of values for the sample mean that would lead to rejecting the null hypothesis. The critical value(s) is (are) the boundary of the critical region. In other words, the critical value is the threshold for xˉ that leads to a p value exactly equal to the chosen significance level.
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