Chapter 7 Lecture 3 Section: 7.5

Estimating a Population Variance Many real situations, such as quality control in a manufacturing process, require that we estimate values of population variances or standard deviations. In addition to making products with measurements yielding a desired mean, the manufacturer must make products of consistent quality that do not fluctuate from extremely good to extremely poor. This consistency can often be measured by the variance or standard deviation. This is vital in maintaining the quality of products and services for the reason that variance and standard deviation measures error. So, the smaller the variance or standard deviation, the smaller the error, the better the quality of the product or service.

s2 is the best point estimate of the population variance σ2. s is a point estimate of σ; however, take into consideration that s is a biased estimator. We will first make our assumptions: 1. The sample is a simple random sample. 2. The population must have normally distributed values. The assumption of a normally distributed population is more critical here. For the methods of this section, departures from normal distributions can lead to unpleasant errors. When developing estimates of variances or standard deviations, we use another distribution, referred to as the chi-square distribution.

Chi–Squared Distribution: 1. The chi-square distribution is not symmetric. 2. As the number of degrees of freedom increases, the distribution becomes more symmetric. 3. The values of chi-square are greater than or equal to 0 4. The chi-square distribution is different for each number of degrees of freedom, given by df = n – 1.

These Critical Values come from Table A-4 The formulas for the confidence intervals are: These Critical Values come from Table A-4 1 – α

Find the critical values that correspond to the given sample size and confidence level. n=80; 99% 2. n=15; 98% 3. n=38; 90% 4. A random sample of size 20 data is taken from a normal population, given below. Find the 95% confidence interval for the population variance. 55 65 58 96 74 85 52 63 87 78 54 57 65 55 85 75 76 74 70 72 By using a calculator, the sample mean is 69.8 with a standard deviation of 12.6.

5. Noise levels at various area urban hospitals were measured in decibels. The noise levels are normally distributed. The mean of the noise levels in 52 corridors was 61.2 decibels, and the standard deviation was 7.9. Find the 99% confidence interval of the true variance. 6. A random sample of 25 car owners results in a mean of 7.01 years and a standard deviation of 3.74 years, respectively. Assuming the sample is drawn from a normally distributed population, find a 90% confidence interval for the population standard deviation.