Estimating Population Parameters 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 DoF … …

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

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 Find the critical values for the following values: 1.95%, n = %, n = %, n = %, n = 70 Find the following confidence intervals for standard deviation 5.95% confidence, n = 15, x- bar = 496, s = % confidence, n = 12, x- bar = $97,334, s = $17, % confidence, n = 25, x- bar = 104, s = % confidence, n = 27, x- bar = 78.8, s = 12.2

More Homework 15 students have cars with a mean worth of $9,500 (s = $2,100) and mean mileage of 27.5 MPG (s = 6.9). 9.Find the interval estimate for value standard deviation 10.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). 11.Find the interval estimate for GPA standard deviation 12.Find the interval estimate for acceptances standard deviation