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

• Χ2l is the left-hand critical value

– Function of the degree of freedom andleft boundary: (1 + degree of confidence) ÷ 2

• Χ2r is the right-hand critical value

– Function of the degree of freedom andright boundary: (1 – degree of confidence) ÷ 2

Reading the table

DoF 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005

…

5 0.412 0.554 0.831 1.145 1.610 9.236 11.071 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

• Χ2r is the right-hand critical value

• Χ2l is the left-hand critical value

2 2

2 2

1 1,

R L

n s n s

Flow

Sample Variance (s2)

Degree of confidence

Sample size (n) Degrees of freedom

Distribution boundaries

Chi-squared table (A-4)2r2L

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 = 30

2. 95%, n = 7

3. 99%, n = 50

4. 90%, n = 70

• Find the following confidence intervals for standard deviation

5. 95% confidence, n = 15, x-bar = 496, s = 108

6. 99% confidence, n = 12, x-bar = $97,334, s = $17,747

7. 90% confidence, n = 25, x-bar = 104, s = 12

8. 99% 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