1. Testing Claims about a Population Standard Deviation
Lesson
Testing Claims about a Population Standard Deviation

2. Objectives
Test a claim about a population standard deviation

3. Vocabulary
Hypothesis – a statement or claim regarding a characteristic of one or more populations
Hypothesis Testing – procedure, base on sample evidence and probability, used to test hypotheses
Null Hypothesis – H0, is a statement to be tested; assumed to be true until evidence indicates otherwise
Alternative Hypothesis – H1, is a claim to be tested.(what we will test to see if evidence supports the possibility)
Level of Significance – probability of making a Type I error, α

4. Chi-square Distribution
Although the chi-square distribution is not symmetric, the basic method of finding critical values is the same as for the normal distribution and the Student’s t-distribution
For example, to compute the critical values for two-tailed tests, with significance level α, we need to find the values for which
The probability is α/2 to the left and
The probability is α/2 to the right

5. Hypothesis Testing (Classical or P-Value) of σ
P-Value is the area highlighted
P-value = P(χ2 < χ2 0)
P-value = P(χ2 > χ2 0)
χ2α/2
χ21-α
χ21-α/2
χ2α
Critical Region
(n – 1)s2
Test Statistic: χ20 =
σ20
Reject null hypothesis, if
P-value < α
Left-Tailed
Two-Tailed
Right-Tailed
χ20 < χ21-α
χ20 > χ2α/2 or χ20 < χ21-α/2
χ20 > χ2α

6. Differences in Tests
One aspect that is different between tests for means and proportions and tests for standard deviations is that tests for standard deviations are usually right-tailed tests
A “do not reject” result – to verify that the standard deviation is low enough (for example for quality control)
A reject result – to verify whether the standard deviation is high enough for significance (which is used in chapters 12, 13, and 14)

7. Notes For two-tailed tests use classical approach
These methods are not robust (if data analysis indicates non-normal population, these procedures are not valid)
Check with Boxplot
No outliers
Test statistic requires squared values
No calculator test available
Χ²cdf(LB, UB, df) gives p-value

8. Example 1
A manufacturer of precision machine parts claims that that the standard deviation of their diameters is mm or less. We perform an experiment on a sample of size, n = 50. We find that our sample standard deviation is s = and the data appears to be bell shaped. At α = 0.01, does the data show significantly that the manufacturer is not meeting their claim? Ho : Ha:
σ =
σ >

9. Example 1 χ249,0.01 = 76.154 (from table VI) 49(0.0012)² (n – 1)s2
= =
0.0010²
(n – 1)s2
Test Statistic: χ20 =
σ20
χ249,0.01 = (from table VI)
Since our test statistic is less than the critical value, we do not reject the null hypothesis. We do not have sufficient evidence to show that the manufacturer is not meeting the objective.

10. Summary and Homework Summary Homework
We can perform hypothesis tests of standard deviations in similar ways as hypothesis tests of means and proportions
Two-tailed, left-tailed, and right-tailed tests
The chi-square distribution should be used to compute the critical values for this test
No calculator help on this test
Homework
pg 556 – 558; 2, 3, 5, 7, 17