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Title: Business Statistics - QBM117


1
Business Statistics - QBM117
  • The p value

2
Objectives
  • Calculation of the p-value of a hypothesis test.
  • Interpretation of the p- value.
  • Testing a hypothesis using the p-value.

3
What is the p-value of a test?
One of the main problems with the testing
procedure used so far is that if we change the
level of significance, this may change the tests
conclusion.
One way of avoiding this is by reporting the
p-value of the test.
The p-value of a test is the smallest value of ?
that would lead to rejection of the null
hypothesis.
It is a measure of the likelihood of obtaining a
particular sample mean or sample proportion, if
the null hypothesis is true.
4
How do we calculate the p-value of a test?
The p-value for a one tail test is found by
for an upper tail test.
p-value
0
z
5
How do we calculate the p-value of a test?
The p-value for a one tail test is found by
for a lower tail test.
p-value
0
z
6
The p-value for a two tailed test is found by
1/2 p-value
1/2 p-value
0
z
7
Exercise 10.19 p342 (9.19 p308 abridged)
p-value
0
z
2.63
8
Exercise 10.18 p342 (9.18 p308 abridged)
1/2 p-value
z
0
-1.76
9
Interpretation of the p-value
The p-value is very important because it measures
the amount of statistical evidence that supports
the alternative hypothesis.
For a given hypothesis test, as the sample
statistic moves further away from the
hypothesised population parameter, the test
statistic gets larger and the p-value gets
smaller.
ie there is more evidence to indicate that the
alternative hypothesis is true.
10
Using the p-value to test hypotheses
A small p-value indicates that if the null
hypothesis is true, then the probability of
obtaining a sample result as extreme as that
obtained is very small ie highly unlikely.
Therefore small p-values lead to rejection of the
null hypothesis.
A large p-value indicates that if the null
hypothesis is true, then the probability of
obtaining a sample result as extreme as that
obtained is reasonably high ie quite likely.
Therefore large p-values lead to non-rejection of
the null hypothesis.
11
What is considered to be a small p-value?
This depends on the researcher however
If the p-value lt ?? it would be considered small
and hence the researcher would reject H0 The
researcher would then conclude there is
sufficient evidence to support the alternative
hypothesis.
If the p-value gt ?? it would not be considered
small and hence the researcher would not reject
H0 . The researcher would then conclude there is
insufficient evidence to support the alternative
hypothesis.
12
Exercise 10.70 p360 (9.70 p328 abridged)
Step 1
Step 2
Step 3
13
Region of non-rejection
0.95
z
0
a 0.05
Critical value
1.645
Step 4
14
Step 5
Step 6
Since 2.9 gt 1.645 we reject H0.
Since 0.0019 lt 0.05 we reject H0.
There is sufficient evidence at ? 0.05 to
conclude that the campaign was a success.
15
How do we calculate the p-value of a test when
our test statistic is a t statistic?
The p-value for a one tail test is found by
for an upper tail test.
p-value
0
t
However the process for calculating this is not
so simple.
16
The p-value for a one tail test is found by
for a lower tail test.
p-value
0
t
17
Example
Exercise 10.26 p346 (9.26 p312 abridged)
You will remember that the hypotheses we were
testing were
The test statistic for this test was t -3.87
t
0
-3.87
18
Since n 15, the degrees of freedom here are 14.
From Table 4, appendix C we find
0.005
2.977
0
19
Therefore due to the symmetric nature of the t
distribution
0.005
-2.977
0
The p-value we require is
Therefore the best we can say is that the p-value
lt 0.005
20
Exercises to be completed before next lecture
SS 10.19 10.21 10.25 10.53 10.58 10.67
(9.19 9.21 9.25 9.53 9.58 9.67
abridged)
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