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
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