Chapter 7 Lecture 3 Section: 7.5

Estimating a Population Variance Many real situations, such as quality control, 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. Qualities that that do not fluctuate in extremes. This consistency can often be measured by the variance or standard deviation. This is vital in maintaining the quality of products and services. Variances and standard deviations measure 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 crucial 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. Table A-4, 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.

Critical Values come from Table A-4 The formulas for the confidence intervals are: 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; 95% 0.005 0.025 0.005 0.025 0.99 0.95

3. A simple random sample of size 20 is taken from a normally distributed population, given below. Find the 98% 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 12.602. df = 19 0.01 0.01 0.98

4. The new GC6 cell phone battery life is normally distributed 4. The new GC6 cell phone battery life is normally distributed. The mean life of 52 randomly selected cell phones batteries is 73.2 hours with a variance of 141.611. Note that these statistics are in reference to batteries that had a full charge and placed on phones and used until the battery ran out. Find the 90% confidence interval of the population standard deviation.