Sec.11.3.ADA

1 Section 11.3 : Testing a single Variance or Standard Deviation Testing σ 2 Theorem 11.1 : If we have a normal population with variance σ 2 and a random sample of n measurements is taken from this population with sample variance s 2 , then χ 2 = ( n − 1 ) s 2 σ 2 i s a value of a random variable having a chi- square distribution with degrees of freedom d .f . = n − 1 . P-values for Chi-Square Distribution Table Two-Tailed test Remember that the P-value is the probability of getting a test statistic as extreme as, or more extreme than, the test statistic computed from the sample. Be sure to choose the area in the appropriate tail (left or right) so that P − value 2 ≤ 0.5 Right-Tail Test Since the Chi-square table gives right-tail probabilities, you can use the table directly to find or estimate the P -value. Pvalue = 2 ndvars ⟶ χ 2 cdf ( χ 2 , 10,000 , Left-Tail Test Since the chi-square table gives right-tail probabilities, you first find or estimate the quantity 1 − ( P − value ) right tail. Then subtract from1 to get the P -value of the left tail. Pvalue = 2 ndVars ⟶ χ 2 cdf ( − 10,000 , χ How to test σ 2 Requirements -You first need to know that a random variable x has a normal distribution. In testing σ 2 , the normal assumption must be strictly observed (whereas in testing means, we can say “normal” or “approximately normal”). Next you need a random sample ( n≥ 2 ) of values from the x distribution for which you compute the sample variance s 2 . 1. In the context of the problem, state the null hypothesis H 0 and the alternate hypothesis H 1 , and set the level of significance α . { H 0 : σ 2 = k H 1 : σ 2 ≠k { H 0 : σ 2 = k H 1 : σ 2 < k { H 0 : σ 2 = k H 1 : σ 2 > k

2 2. Use the value of σ 2 given in the null hypothesis H 0 , the sample variance s 2 , and the sample size n to compute the χ 2 value of the sample test statistic s 2 ; χ 2 = ( n − 1 ) s 2 σ 2 with degrees of freedom d .f . = n − 1 . 3. Use a chi-square distribution and the type of test to find or estimate the P -value. 4. Conclude the test. If P − value ≤α , then reject H 0 , if P − value > α , then do not reject H 0 . 5. Interpret your conclusion in context of the application. Example 1 : For years, a large discount store has used independent lines to check out customers. Historically, the standard deviation of waiting times is 7 minutes. The manager tried a new, single-line procedure. A random sample of 25 customers using the single-line procedure was monitored, and it was found that the standard deviation for waiting times was only s = 5 minutes. Use α = 0.05 to test the claim that the variance in waiting times is reduced for the single-line method. Assume the waiting times are normally distributed. Example 2 : Let x be a random variable that represents weight loss (in pounds) after following a certain diet for 6 months. After extensive study, it is found that x has a normal distribution with σ = 5.7 pounds. A new modification of the diets has been implemented. A random sample of n = 21 people use the modified diet for 6 months. For these people, the sample standard deviation of weight loss is s = 4.1 pounds. Does this result indicate that the variance of weight loss for the modified diet is different (either way) from the variance of weight loss for the original diet? Use α = 0.01 . Assume weight loss for each diet follows a normal distribution.