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Estimating Population Variance and Chi-square Distribution Properties

Learn about estimating population variance, constructing confidence intervals, and properties of Chi-square distribution with examples and exercises to enhance your understanding.

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Estimating Population Variance and Chi-square Distribution Properties

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  1. 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 ???? R L R L

  5. Our Distributions 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.

  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. 377-379 • #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. • 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.

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