Testing Claims about a Population Standard Deviation Lesson 10 - 5 Testing Claims about a Population Standard Deviation

Objectives Test a claim about a population standard deviation

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

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

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

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)

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

Example 1 A manufacturer of precision machine parts claims that that the standard deviation of their diameters is .0010 mm or less. We perform an experiment on a sample of size, n = 50. We find that our sample standard deviation is s = 0.0012 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 = 0.0010 σ > 0.0010

Example 1 χ249,0.01 = 76.154 (from table VI) 49(0.0012)² (n – 1)s2 = ---------------- = 70.56 0.0010² (n – 1)s2 Test Statistic: χ20 = ------------- σ20 χ249,0.01 = 76.154 (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.

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