Estimating a Population Variance Section 7.5 Estimating a Population Variance

Symbol Check = population mean = population standard deviation = sample standard deviation = sample mean = population variance = sample variance

Estimators of The sample variance is the best point estimate of the population variance . The sample standard deviation s is commonly used as a point estimate of (even though it is a biased estimate).

Topic Preview – Sneak Peak Constructing Confidence Intervals Requirements 1.) The sample is a simple random sample. 2. The population must have a normal distribution (even if sample is large). Confidence Interval for the Population Variance Confidence Interval for the Population Standard Deviation ???? R L R L

Student t Distribution Chi-square Distribution Our Distributions Normal Distribution Student t Distribution Chi-square Distribution Estimates of proportions or means with known Estimates of means with known s. Estimates of variance or standard deviations. NEW!

Properties of the Chi-square Distribution Uses Table A4 To find your Chi-square distribution value you must know: Degrees of Freedom ( df = n-1) Area located to the right of the critical value.

Properties of the Chi-square Distribution Chi-square is NOT symmetric However, the distribution becomes more symmetric as the degrees of freedom increase Interval Notation: (s²-E < σ <s²+E) Interval Notation: (s²-E, s²+E) Interval Notation: (s²±E)

Using Table A4 Find for the population standard deviation σ with a confidence level of 95% and a sample size of n=10.

Constructing a Confidence Interval Twelve different video games showing substance use were observed and the duration of times of game play (in seconds) are listed below. Use the sample data to construct a 99% confidence interval estimate of σ, the variance and standard deviation of the duration of game play. 4049 3884 3859 4027 4318 4813 4657 4033 5004 4823 4334 4317

Determining Sample Size We want to be 95% confident that our estimate is within 20% of the true value of σ. Find the sample size.

Homework Pg. 377-379 #2, 6, 7, 16, 19

Homework Answers: Pg. 377-379#2, 6, 7, 16, 19 2. Yes, no. The format s +/- E cannot be used due to the lack of symmetry of chi-square distribution. The confidence interval does not have s at the center. 6. 8.907, 32.852 7.) 51.172, 116.321 16. 211 19. a.) 17.7 min < < 32.4 min. b.) 14.7 min < < 35.3 min c.) The two confidence intervals are not substantially different. There does not appear to be a difference between the standard deviation of lengths of movies rated PG or PG-13 and the lengths of movies rated R.