P-value for a Mean
v0.99
October 15, 2015
Theory: The P-value for a Data-Set-Given-a-Hypothesis
Suppose you have data in the form of a sample (a list) of n = 10 numbers drawn/selected/chosen
(randomly) from an idealized population of oblongated yellow widgets.
Data: 20.5, 16.9, 19.2, 22.4, 19.6, 24, 21.2, 22.9, 24, 22.3
You compute the mean and standard deviation.
The Data’s summary statistics: x̄ = 21.30, and s = 2.28.
It has long been believed that the average value µ expected for a randomly selected inverted yellow widgets was exactly 20. That is, it has been believed that µ = 20.
So, x̄ = 21.30 and µ = 20.
Houston, do we have a problem?
To decide whether the data is consistent with the belief, we compute the Probability-value
(the P-value) of the data, given the belief.
We Compute:
the Probability-Value of the data given the belief µ = 20
at the 5% significance level.
s
Step 1. Calculate the Standard Error for the sample mean for samples of size n: SE = √ .
n
2.28
2.28
≈ 0.7200.
SE = √ ≈
3.16
10
Step 2. Calculate the test statistic using t =
x̄ − µ
.
SE
21.30 − 20
1.30
=
= 1.806.
0.7200
0.7200
Step 3. Using the t-distribution calculator at Statkey with the degrees of freedom
df = 10 − 1 = 9, we use the t-distribution tail probability calculator, to find the
P -value of t = 1.806 .
t
=
We get the P -value of t = 1.806 is
P = 0.052 + 0.052 = 0.104 = 10.4%.
Randomization: The P-value for a Data-Set-Given-a-Hypothesis
Step 1. Enter the data and the Hypothesis, µ = 20.
Step 2. Generate the Randomization.
Step 3. We get the P -value of x̄ = 21.3 is
P = 0.015 + 0.015 = 0.030 = 3.0%.