1. Estimating a Population Variance
Section 7.5
Estimating a Population Variance

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

3. 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).

4. 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

5. 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!

6. 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.

7. 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)

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

9. 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

10. 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.

11. Homework Pg
#2, 6, 7, 16, 19

12. 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.
, ) ,
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.