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Elementary Statistics Triola, Elementary Statistics 11/e Unit 16 The Confidence Interval for Variation Estimation Unit 16 The Confidence Interval for Variation (Section 7-5) The sampling distribution of the population variance, ππ is the Chi Square distribution. This is a new probability distribution for us. It is pictured below, There are a few things to notice about this distribution. The most obvious is that it is not symmetrical, but rather, skewed to the right. Another interesting point is that the left tail ends at zero. There are no negative values along the π 2 axis. Finally, similar to the Student t distribution, the exact shape of the Chi Square distribution depends on the degrees of freedom which equal one less than the sample size. The symbol for the Chi Square variable is π 2 . This is not X squared. The superscript is just part of the notation. Because the distribution is not symmetrical, we canβt come up with interval expression like π 2 ± πΈ. Instead, we find a left critical value, ππΏ2 that has an area under the curve to the left of it equal to πΌβ and a right critical value, π 2 , that has an area under the curve to the right of it equal to πΌβ . See π 2 2 the figure below, This figure illustrates the situation when we are calculating a 95% confidence interval for a sample of size 10. The degrees of freedom is 9 and the areas that are shaded red sum to πΌ = 0.05. There is no margin of error when calculating confidence intervals for variances. There is only an upper bound and a lower bound as shown by the following expression for the interval, 57 Elementary Statistics Triola, Elementary Statistics 11/e Unit 16 The Confidence Interval for Variation (π β 1)π 2 (π β 1)π 2 β β < π < ππ 2 ππΏ2 Notice a few things here. First, the only information that you need from the sample is its size and its standard deviation, s. Also notice that ππ 2 is on the left side of the expression and ππΏ2 is on the right. This makes sense since they are in the denominators and ππ 2 > ππΏ2 . When we calculated the confidence interval for means, we used T.INV.2T to find the critical value. This Excel tool is a two-tail tool and we only need one critical value, π‘πΌβ2 because the Student t distribution is symmetrical. This is not the case here. We will need to find two critical value, ππ 2 and ππΏ2 , and we will need two separate Excel tools to do so, CHISQ.INV.RT and CHISQ.INV respectively. Remember, variation, π 2 , is simple the square of standard deviation, π. Worked Example Letβs say that we want to estimate variation of the weight of U.S. quarters. We collect a simple random sample of 30 quarters, weigh them and calculate the standard deviation of the sample, π = 4.7 gm. Itβs our goal to calculate the 95% confidence interval for the population standard deviaiton, π. πΌ πΌ We will need to calcualte ππΏ2 . The area to the left of ππΏ2 is 2 where πΌ = 1.0 β .095 = 0.05. Hence , 2 = 0.05. We use CHISQ.INV to find ππΏ2 , 58 Elementary Statistics Triola, Elementary Statistics 11/e Unit 16 The Confidence Interval for Variation In a similar fashion, we find ππ 2 , Here is an excel spreadsheet for this example. If you double click on the cells below, you will open Excel and you can explore the contents of the cells. s n 4.7 XL XR LB UB 30 16.04707 45.72229 3.743113 6.318281 The confidence interval is thus given by, 3.7431 < π < 6.3183 This is the end of Unit 16. In class, you will get more practice with these concepts by working exercises in MyMathLab. 59