Transcript Z - Wsfcs

Tests of Significance
-how to put them in writing
First we list the steps:
Step 1:
Null and alternate hypotheses
Step 2:
Conditions
Step 3:
Formula and calculations
Step 4:
Graph of Z
Step 5:
P-value
Step 6:
Compare p and α, then reject/fail to reject
Step 7:
Conclusion in context
We consider the problem of finding whether the average weight
of an ICR white laboratory mouse is the value we believe it to
be. We read that the mean and standard deviation are 37g and
3.2g, respectively. We collect a sample of 10 mice and record
these weights (in grams): {35, 42, 41, 34, 38, 36, 37, 39, 34,
36}. Like many other measurements, the weights of the mice
follow a normal distribution. The question we are asked is
whether the mean weight of the population of all mice in our
university laboratories is 37g, the same as the known value for
ICR whites. Carry out a test of significance at α = 0.05.
Now we will write an answer to this question, following the 7
steps.
We give more detail as we work:
Step 1:
Write the null and alternate hypotheses in
symbols.
H0: μ = 37g
Ha: μ ≠ 37g
Step 2:
Conditions: meet every condition
required for the test.
We are given an SRS, and a normal population.
Step 3:
Formula and calculations: Write the formula
for the calculation of the test statistic (Z) and
show the values substituted into the equation.
Solve for Z.
Z=
x - m0
37.2 - 37
.2
s = 3.2 = 1.01192 » .1976
n
10
n.b., 37.2 was found by calculating the mean of the data.
Step 4:
Graph the test statistic.
Step 5:
Calculate the probability of the shaded region
of the graph. Write a statement using Z.
P-value = P(Z £ -.1976 or Z ³ .1976) =
= 2(.42166) » .8433
Step 6:
Compare to alpha, then reject or fail to reject H0.
Fail to reject H0, a test statistic this extreme
will occur by chance alone 84% of the time.
n. b., if you have trouble deciding whether to reject, ask
yourself if the result is surprising? Are you surprised that we
could get a mean of 37.2 when the true mean is 37? I’m not!
Step 7:
Write a conclusion in terms of the original
problem. I usually start with “We have
evidence of…” or “We lack evidence of…,”
and modify appropriately.
We lack evidence that the mean weight of our
university laboratory mice is different from 37
grams.
n.b., We either reject or fail to reject, but we never attempt to
prove that the null hypothesis is true!
We will have many variations, but the basic process will be the
same now for all tests of significance.
Remember the 7 steps:
Step 1:
Null and alternate hypotheses
Step 2:
Conditions
Step 3:
Formula and calculations
Step 4:
Graph
Step 5:
P-value
Step 6:
Reject/fail to reject and explain P-value.
Step 7:
Conclusion
THE END