One-Sample Tests for Population Means
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One-Sample Tests for Population Means |
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Introduction
The aim of these tests is to compare a population mean, 
, to a value, 
, referred here to as the hypothetical mean of the test, based on a random sample, 
, of size 
, drawn from the population in question.
Methods
Three different methods are provided for this test:
| 1. | z-Procedure for Known Population Variance |
For instances where the population variance is known, we use the test statistic
,
where 
is the sample mean, is the hypothetical mean of the test, 
is the square root of the population variance, or population standard deviation, and is the number of observations in the sample.
When the population variance is not known, the test statistic 
has a Student's t-distribution with 
degrees of freedom. However, if the sample's size is sufficiently large, the square root of the unbiased sample variance, 
, can be appropriate for a z-test on the basis that as the size of a sample increases, its t-distribution approximates the normal distribution. Hence, for large ,
,
where 
and 
are the P-Values of according to the normal distribution and the t-distribution with degrees of freedom, respectively, and is the square root of the unbiased sample variance.
| 3. | t-Procedure for Unknown Population Variance |
If we consider the fact that the reason behind these tests is to infer population parameters from small samples. Then, when the population variance is not known, the procedure of choice should be the t-procedure, which takes the form
,
and has a Student's t-distribution with degrees of freedom.
Hypotheses
The null hypothesis takes the form 
, with the following alternative hypotheses:
| § |  for two-tailed tests, |
| § |  for left-tailed tests, and |
| § |  for right-tailed tests. |
Refer to Alternative Hypothesis for additional information of this subject.
One-Sample z-Test for Means (Known Variance)
Sample: |
<The Sample Number> |
Observations: |
<Observations in the Sample> |
Hypothetical Mean: |
<The Hypothetical Mean> |
Significance Level: |
<The Significance Level> |
Mean: |
<The Sample Mean> |
Population Variance: |
<The Population Variance> |
z: |
<z> |
One-Tailed P(|z|): |
<The One-Tailed P-Value of |z| > |
One-Tailed z Critical (+/-): |
<The One-Tailed Absolute Critical Point> |
Left Tailed-Test: |
"[Do not ]Reject Ho" |
Right-Tailed Test: |
"[Do not ]Reject Ho" |
Two-Tailed 2P(|z|): |
<The Two-Tailed P-Value of |z|, doubled > |
Two-Tailed z Critical (+/-): |
<The Two-Tailed Absolute Critical Point> |
Two-Tailed Test: |
"[Do not ]Reject Ho" |
One-Sample z-Test for Means (Unknown Variance)
Sample: |
<The Sample Number> |
Observations: |
<Observations in the Sample> |
Hypothetical Mean: |
<The Hypothetical Mean> |
Significance Level: |
<The Significance Level> |
Mean: |
<The Sample Mean> |
Sample Variance: |
<The Sample Variance> |
z: |
<z> |
One-Tailed P(|z|): |
<The One-Tailed P-Value of |z| > |
One-Tailed z Critical (+/-): |
<The One-Tailed Absolute Critical Point> |
Left Tailed-Test: |
"[Do not ]Reject Ho" |
Right-Tailed Test: |
"[Do not ]Reject Ho" |
Two-Tailed 2P(|z|): |
<The Two-Tailed P-Value of |z|, doubled > |
Two-Tailed z Critical (+/-): |
<The Two-Tailed Absolute Critical Point> |
Two-Tailed Test: |
"[Do not ]Reject Ho" |
One-Sample t-Test for Means
Sample: |
<The Sample Number> |
Observations: |
<Observations in the Sample> |
Hypothetical Mean: |
<The Hypothetical Mean> |
Significance Level: |
<The Significance Level> |
Mean: |
<The Sample Mean> |
Sample Variance: |
<The Sample Variance> |
t: |
<t> |
Degrees of Freedom: |
<Number of Degrees of Freedom> |
One-Tailed P(|t|): |
<The One-Tailed P-Value of |t| > |
One-Tailed t Critical (+/-): |
<The One-Tailed Absolute Critical Point> |
Left Tailed-Test: |
"[Do not ]Reject Ho" |
Right-Tailed Test: |
"[Do not ]Reject Ho" |
Two-Tailed 2P(|t|): |
<The Two-Tailed P-Value of |t|, doubled > |
Two-Tailed t Critical (+/-): |
<The Two-Tailed Absolute Critical Point> |
Two-Tailed Test: |
"[Do not ]Reject Ho" |