1. IE 355: Quality and Applied Statistics I Confidence Intervals
IENG Lecture 08
Confidence Intervals
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

2. IE 355: Quality and Applied Statistics I
Assignment:
Preparation:
Print Hypothesis Test Tables from Materials page
Have this available in class …or exam!
Reading:
Chapter 3:
3-1 through 3-1.1, 3-2 through 3-3.4, 3-4 through 3-4.3
HW 2:
CH 3: # 1, 5, 9, 11, 17, 18, 21
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

3. Estimation of Process Parameters
IE 355: Quality and Applied Statistics I
Estimation of Process Parameters
In SPC:
the probability distribution is used to model a quality characteristic (e.g. dimension of a part, viscosity of a fluid)
Therefore:
we are interested in making inferences about the parameters of the probability distribution (e.g. mean m and variance s2)
Since:
Values of parameters are generally not known, we need to estimate them from sample data
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

4. IE 355: Quality and Applied Statistics I
Point Estimate
Numerical value, computed from a sample of data, used to estimate a parameter of a distribution
Example:
Say we take n = 50 measurements of a quality characteristic
Sample mean is point estimate of m i.e.
Sample variance is point estimate of s2 i.e.
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

5. IE 355: Quality & Applied Statistics I
Confidence Intervals
A confidence interval for an unknown parameter  is an interval that contains a set of plausible values of the parameter. It is associated with a confidence level
1- , which measures the probability that the confidence interval actually contains the unknown parameter.
4/12/2019
IE 355: Quality & Applied Statistics I

6. Confidence Interval (C.I.) (Interval Estimate)
IE 355: Quality and Applied Statistics I
Confidence Interval (C.I.) (Interval Estimate)
A C.I. is an interval that, with some probability, includes the true value of the parameter
Ex. C.I. of mean m is 
L - lower confidence limit
U - upper confidence limit
(1-a) - probability that true value of parameter lies in interval (we pick a)
The interval L  m  U is called a 100(1-a)% C.I. for the mean m
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

7. C.I. on the Mean of Normal Distribution with Variance Unknown
IE 355: Quality and Applied Statistics I
C.I. on the Mean of Normal Distribution with Variance Unknown
Suppose , and
We don't know the true mean m or true variance s2
A 100(1-a)% C.I. for the unknown (true) mean m is:
- sample mean
S - sample standard deviation
n - number of observations in sample
- value of t distribution
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

8. Ex. C.I. on the Mean of Normal Distribution with Variance Unknown
IE 355: Quality and Applied Statistics I
Ex. C.I. on the Mean of Normal Distribution with Variance Unknown
Automatic filler deposits liquid in a container.
WANT: 95% C.I. on the mean amount (ounces) per container
Collect random sample: x1, x2, …, xn say n = 10
Compute sample average:
Compute sample standard deviation (square root of variance):
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

9. IE 355: Quality and Applied Statistics I
Ex. C.I. on Mean cont'd
Find the t-distribution value:
Look in Table (Appendix IV)
Want a 95% C.I. so, 100(1-a)% = 95%  a = 0.05  = degrees of freedom = (n-1) = 9 so …
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

10. IE 355: Quality & Applied Statistics I
4/12/2019
IE 355: Quality & Applied Statistics I

11. IE 355: Quality and Applied Statistics I
Ex. C.I. on Mean cont'd
Find the t-distribution value:
Look in Table (Appendix IV)
Want a 95% C.I. so, 100(1-a)% = 95%  a = 0.05  = degrees of freedom = (n-1) = 9 so …
Substitute into C.I.
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

12. Interpretation of a 95% C.I.
IE 355: Quality and Applied Statistics I
Interpretation of a 95% C.I.
Repeat sampling 10,000 (or many, many) times & obtain C.I.s
Each C.I. will have (slightly) different center point and width
On average, 95% of the C.I.s will include the true mean
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

13. C.I.s on Other Parameters and Quantities
IE 355: Quality and Applied Statistics I
C.I.s on Other Parameters and Quantities
Same procedure, different formulas
For example, C.I. on
Mean (of any distribution) when variance is known
Variance of a normal distribution
Difference in two means (of any distribution) when variances are known
Difference in two means from normal distribution when variances are unknown
Ratio of variances of two normal distributions
etc. ...
(See textbook Sections 3-3, 3-4 to review)
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

14. Confidence Intervals – Single Mean
IE 355: Quality and Applied Statistics I
Confidence Intervals – Single Mean
Two-Sided t-Interval: A confidence interval with confidence level 1-  for a population mean µ based upon a sample of n continuous data observations with a sample mean x and a sample standard deviation s is
This is known as a two-sided confidence-interval
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

15. Confidence Intervals – Two Means
IE 355: Quality and Applied Statistics I
Confidence Intervals – Two Means
Two-Sample t-interval (Equal Variances): Consider a sample of size n from a population A with sample mean x, and a sample standard deviation sx, and a sample of size m from a population B with a sample mean y, and a sample standard deviation sy. If an experimenter assumes that the population variances 2A and 2B are equal, then a two-sided 1-  confidence interval for the difference in population means µA - µB is: With the pooled variance (sp2) estimated as:
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

16. Confidence Intervals – Two Means
IE 355: Quality and Applied Statistics I
Confidence Intervals – Two Means
Two-Sample t-Procedure (Unequal Variances): Consider a sample of size n from a population A with sample mean x, and a sample standard deviation sx, and a sample of size m from a population B with a sample mean y, and a sample standard deviation sy. A two-sided 1- confidence interval for the difference in populations means µA - µB is where the degrees of freedom of the critical point are
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

17. Confidence Intervals – One Variance
IE 355: Quality and Applied Statistics I
Confidence Intervals – One Variance
Confidence Interval for 2: If s2 is the variance of a random sample of size n from a normal population, a (1-)100% confidence interval for 2 is given by: where and are with n-1 degrees of freedom, leaving areas of /2 on right and 1-/2 to the left.
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

18. Single Sample: Variances
IE 355: Quality and Applied Statistics I
Single Sample: Variances
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl

19. Confidence Intervals - Two Variances
IE 355: Quality and Applied Statistics I
Confidence Intervals - Two Variances
Confidence Interval for 2 / 2 : If s12 and s22 are the variances of random samples of size n1 and n2 both from normal populations, then a (1-)100% confidence interval for 12/ 22 is:
where f/2 (1 ,2) is an f-value with 1=n1-1 and 2=n2-1 degrees of freedom, leaving an area of /2 to the right, and f/2 (2 ,1) is a similar f-value with 2=n1-1 and 1=n2-1 degrees of freedom.
4/12/2019
IE 355: Quality & Applied Statistics I
(c) 2002, D.H. Jensen & R. C. Wurl