Download 6-6 variance interval estimate

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 and
left boundary: (1 + degree of confidence) ÷ 2
• Χ2r is the right-hand critical value
– Function of the degree of freedom and
right boundary: (1 – degree of confidence) ÷ 2
Reading the table
DoF 0.995
0.99
0.975
0.95
0.90
0.10
0.05
0.554
0.831
1.145
1.610
9.236
11.071
0.025
0.01
0.005
…
5
…
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  1 s

•
•
•
•
2
R
2
,
 n  1 s

2
L
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
Flow
Sample Variance (s2)
Sample size (n)
Interval estimate
Degrees of freedom
Chi-squared table (A-4)
Critical values


2
r
2
L
Degree of confidence
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