Estimating Population Parameters Mean Variance and standard deviation

Estimating Population Parameters ü Mean • Variance (and standard deviation) – Degrees of Freedom • Sample size – 1 – Sample standard deviation – Degrees of confidence (e. g. , 95%) • Proportion

Critical value • Boundary between what’s in and what’s outside the confidence interval • Intermediate value in the calculation of the margin of error • Mean and proportion – Samples statistics are normally distributed – Use the z table or t (Student) table • Variances – Samples variances are not normally distributed – Use the Chi-square distribution table (A-4)

Critical Value for Mean and Proportion • Critical value is the confidence interval on the standard normal distribution • Margin or error calculation maps the critical value to a range appropriate for our data, after adjusting for the sample size

Critical Value for Variance and Standard Deviation • • Chi-squared (χ2) Distribution The distribution of sample variances Skewed to the right Shape varies according to degrees of freedom

Boundaries • Χ 2 l is the left-hand critical value – Function of the degree of freedom and left boundary: (1 + degree of confidence) ÷ 2 • Χ 2 r is the right-hand critical value – Function of the degree of freedom and right boundary: (1 – degree of confidence) ÷ 2

Reading the table Do. F 0. 995 0. 99 0. 975 0. 90 0. 10 0. 05 0. 554 0. 831 1. 145 1. 610 9. 236 11. 071 0. 025 0. 01 0. 005 … 0. 412 12. 833 15. 086 16. 750

Finally, the calculation • We do not calculate a margin of error, but rather the upper and lower boundaries directly: • • n is the number of samples s is the sample’s standard deviation Χ 2 r is the right-hand critical value Χ 2 l is the left-hand critical value

Flow Sample Variance (s 2) Sample size (n) Interval estimate Degrees of freedom Chi-squared table (A-4) Degree of confidence Critical values Distribution boundaries

For example • A random sample of 25 students has a mean math SAT score of 560 with a standard deviation of 50 points. What is 90% confidence interval for the population standard deviation? • Degrees of freedom: • Degree of confidence: • Left boundary: • Left-hand critical value • Right boundary: • Right-hand critical value

Live example • A random sample of 60 cars has a mean gas mileage of 22 MPG with a standard deviation of 6 MPG points. What is 95% confidence interval for the population standard deviation? • Degrees of freedom: • Degree of confidence: • Left boundary: • Left-hand critical value • Right boundary: • Right-hand critical value

Your turn • A random sample of 80 bowlers has a mean score of 145 with a standard deviation of 45 pins. What is 95% confidence interval for the population standard deviation?

Homework • 1. 2. 3. 4. Find the critical values for the following values: 95%, n = 30 95%, n = 7 99%, n = 50 90%, n = 70 • 5. 6. 7. 8. Find the following confidence intervals for standard deviation 95% confidence, n = 15, xbar = 496, s = 108 99% confidence, n = 12, xbar = $97, 334, s = $17, 747 90% confidence, n = 25, xbar = 104, s = 12 99% confidence, n = 27, xbar = 78. 8, s = 12. 2

More Homework • 9. 10. • 11. 12. 15 students have cars with a mean worth of $9, 500 (s = $2, 100) and mean mileage of 27. 5 MPG (s = 6. 9). Find the interval estimate for value standard deviation Find the interval estimate for MPG standard deviation 8 seniors have an mean GPA of 3. 6 (s = 1) and a mean number of college acceptances of 5. 5 (s = 2. 2). Find the interval estimate for GPA standard deviation Find the interval estimate for acceptances standard deviation