392 Chapter 8 Confidence Intervals Check Your Understanding 1. Find the critical values for a 95% confidence interval using the chi-square distribution with 18 degrees of freedom. 8.231, 31.526 2. Find the critical values for a 99% confidence interval using the chi-square distribution with 25 degrees of freedom. 10.520, 46.928 Answers are on page 395. Objective 2 Construct confidence intervals for the variance and standard deviation of a normal distribution Confidence Intervals for the Variance and Standard Deviation We will present the method for constructing confidence intervals for the variance and the standard deviation of a normal distribution. We will follow with an example, then present a justification for the method. Confidence Intervals for the Variance and Standard Deviation Let s2 be the sample variance from a simple random sample of size n from a normal population. A level 100(1 − ��)% confidence interval for the population variance ��2 is (n − 1)s2 ��2 ��∕2 < ��2 < (n − 1)s2 ��2 1−��∕2 Explain It Again Degrees of freedom: When constructing a confidence interval for a variance or standard deviation, the number of degrees of freedom is always 1 less than the sample size. A level 100(1 − ��)% confidence interval for the population standard deviation �� is √ (n − 1)s2 ��2 ��∕2 < �� < √ (n − 1)s2 ��2 1−��∕2 The critical values are taken from a chi-square distribution with n − 1 degrees of freedom. Step-by-step procedure for constructing confidence intervals We summarize the procedure for constructing confidence intervals for the variance and standard deviation of a normal population. Procedure for Constructing Confidence Intervals for the Variance and Standard Deviation of a Normal Distribution Step 1: Compute the sample variance s2, if it isn’t given. Step 2: Find the critical values ��2 1−��∕2 and ��2 ��∕2, using the chi-square distribution with n − 1 degrees of freedom. Step 3: Compute the lower and upper confidence bounds: Lower bound = (n − 1)s2 ��2 ��∕2 Upper bound = (n − 1)s2 ��2 1−��∕2 Step 4: The level 100(1 − ��)% confidence interval for ��2 is (n − 1)s2 ��2 ��∕2 < ��2 < (n − 1)s2 ��2 1−��∕2 To find a level 100(1 − ��)% confidence interval for the population standard deviation ��, take the square roots of the confidence bounds for the variance: √ (n − 1)s2 ��2 ��∕2 < �� < √ (n − 1)s2 ��2 1−��∕2 Step 5: Interpret the result.
navidi_monk_elementary_statistics_2e_ch7-9
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