Statistical Significance
The power of ALPHA
The decisive value of P is called
the significance level. We write it
as α, the Greek letter alpha.
“Significant” in the statistical sense
does not mean “important.” It means
simply “not likely to happen just by
chance.”
Statistical Significance
If the P-value is as small as or
smaller than alpha, we say that the
data are statistically significant at
level α.
In practice, the most commonly used
significance level is: α = 0.05
To test the hypothesis H0: μ= μ0 based on an SRS of size n from a
population with unknown mean μ and known standard deviation
σ, compute the one-sample z statistic
z= x-ℳ
σ/√n
Step 1: Hypotheses Identify the population of interest and the
parameter you want to draw conclusions about. State hypotheses.
Step 2: Conditions Choose the appropriate inference procedure.
Verify the conditions for using it.
Step 3: Calculations If the conditions are met, carry out the
inference procedure.
•Calculate the test statistic.
Find the P-value.
Step 4: Interpretation Interpret your results in the context of the
problem.
•Interpret the P-value or make a decision about H0 using statistical
significance.
Don't forget the 3 C's: conclusion, connection, and context.
reject H0
or
fail to reject H0
we will reject H0 if our result is statistically significant
at the given α level.
That is, we will fail to reject H0 if our result is not
significant at the given α level.
EXAMPLE
Ho: µ = 0, there is NO difference in job satisfaction between the two work environmen
Ho: µ ≠ 0, there is a difference in job satisfaction between the two work environments
α = .05
p = .0234
Therefore, our hypothesis testing for this particular
case is statistically significant at α = .05
A certain random number generator is supposed to produce random
numbers that are uniformly distributed on the interval from 0 to 1.
If this is true, the numbers generated come from a population with
μ = 0.5 and σ = 0.2887. A command to generate 100 random
numbers gives outcomes with mean x = 0.4365. Assume that the
population σ remains fixed. We want to test H0: μ= 0.5 versus Ha: μ
≠ 0.5.
(a) Calculate the value of the z test statistic and the P-value.
(b) Is the result significant at the 5% level (α = 0.05)? Why or why
not?
(c) Is the result significant at the 1% level (α = 0.01)? Why or why
not?
(d) What decision would you make about H0 in part (b)? Part (c)?
Explain.
(a) Calculate the value of the z test statistic and the P-value.
(b) Is the result significant at the 5% level (α = 0.05)? Why or
why not? Since the P-value is less than 0.05, we say
that the result is statistically significant at
the 5% level.
(c) Is the result significant at the 1% level (α = 0.01)? Why or
why not?
Since the P-value is greater than 0.01, we
say that the result is not statistically
significant at the 1% level.
(d) What decision would you make about H0 in part (b)? Part
(c)? Explain.
At the 5% level, we would reject Ho and conclude that
the random number generator does not produce
numbers with an average of 0.5.
At the 1% level, we would not reject Ho and conclude
that the observed deviation from the mean of 0.5 is
something that could happen by chance.
That is, we would conclude that the random number
generator is working fine at the 1% level