AP Statistics
Significance Tests for Means
One Sample Z-Test - for means when the population standard deviation is given.

Called a one sample z test for means; (“Z Test” in calculator)
P – Parameter = μ. Identify the Population when stating parameter. Process – identify the test
A – Assumptions & Conditions
1. Random  SRS/randomized experiment/ nonbias sample
2. Normality 
a. Given raw data, make a graph. If it is a small sample, it should be somewhat
symmetrical and you can proceed with z procedure. If raw data is very skewed and the
sample is smaller than 30, then you cannot proceed with z procedure. If the sample size is
bigger than 30, then you can proceed with the z procedure according to the Central Limit
Theorem.
b.
If not given raw data, use common sense. If the sample is large, the central limit
theorem applies and you may proceed with the procedure. If small, think about whether the
distribution should be normal.
3. Independence  check to see if the population is at least 10 times larger than the sample
by adding a 0 to the sample size and verifying the population size.
S – Statistical Work
Sampling distribution that you use for the calculation has a mean of μ and standard deviation
of σ/√n  N(𝜇, 𝜎/√𝑛)
The test statistic is
𝑧=
𝑥−𝜇
𝜎/√𝑛
Determine p-value from Z using either Table A or calculator with 2nd  VARS  2:
normalcdf(lower bound, upper bound). ***If two-sided, multiply by your answer by 2***
S – Summary
Decision  make a decision based on your p-value and alpha level. “Because the p-value is
less than the alpha level of ________, we can reject the null hypothesis.” OR “Because the p-value is
not less than the alpha level of ________, we fail to reject the null hypothesis.”
Conclusion  when p-value is lower than alpha: “We can conclude that ____________
(state alternative hypothesis).
*When p-value is higher than alpha: “We cannot conclude that ____________(state alternative
hypothesis).
One Sample T-test - for means when the population standard deviation is not given.
P – Parameter = μ. Identify the Population when stating parameter. Process – identify the test
A – Assumptions & Conditions  same as Z-Test (Random, Normal, Independent)
S – Statistical Work
Sampling distribution that you use for the calculation has a mean of μ and standard error of
s/√n  N(𝜇, 𝑠/√𝑛)
The test statistic is
𝑡=
𝑥−𝜇
𝑠/√𝑛
Determine p-value from t using either Table B or calculator with 2nd  VARS  tcdf(lower
bound, upper bound). ***If two-sided, multiply by your answer by 2***
BE SURE TO GIVE DEGREES OF FREEDOM.
S – Summary
Decision  make a decision based on your p-value and alpha level. “Because the p-value is
less than the alpha level of ________, we can reject the null hypothesis.” OR “Because the p-value is
not less than the alpha level of ________, we fail to reject the null hypothesis.”
Conclusion  when p-value is lower than alpha: “We can conclude that ____________
(state alternative hypothesis).
*When p-value is higher than alpha: “We cannot conclude that ____________(state alternative
hypothesis).
Example 1) The environmental protection agency has determined that safe drinking water
should contain no more than 1.3 mg/liter of copper. You are testing water from a new
source, and take 30 water samples. The mean copper content is 1.36mg/liter with a
standard deviation of 0.15. Do these samples provide convincing evidence that the water
from this source contains unsafe levels of copper?
P
A
S
S