Transcript MATH408: PROBABILITY & STATISTICS

MATH408: Probability & Statistics
Summer 1999
WEEK 7
Dr. Srinivas R. Chakravarthy
Professor of Mathematics and Statistics
Kettering University
(GMI Engineering & Management Institute)
Flint, MI 48504-4898
Phone: 810.762.7906
Email: [email protected]
Homepage: www.kettering.edu/~schakrav
Test on population variance
•Recall that sample variance is an UMV for the population variance.
•To test Ho: 2 = 2o vs Ho: 2 > 2o we look at the test statistic :
Decision Rule: Reject Ho if the absolute value of the calculated value of
the above statistic exceeds the table value (2,n-1).
EXAMPLES
EXAMPLES
Inference on Population Proportion
Test of hypotheses
•Recall that sample proportion is an UMV for the population proportion
•To test Ho: p = po vs Ho: p  po we look at the test statistic
Decision Rule: Reject Ho if the absolute value of the calculated value of
the above statistic exceeds the table value (z/2).
EXAMPLES
Inference on Population Proportion
Confidence Interval
EXAMPLES
CHOICE OF SAMPLE SIZE
EXAMPLES
HOMEWORK PROBLEMS
Sections 4.6 through 4.7
43-46, 48-50, 52, 54-56, 59
Decision Making for Two Samples
• So far we talked about making inferences about
the population parameter(s) when dealing with
only one population at a time.
• Suppose we ask:
– Is the new method of assembling a product better than
the existing one?
– Is there any difference between workers in two
assembly plant?
– How do we answer these scientifically?
Using statistical methods for two-population case.
Inference for a difference in the means
Under the assumption listed earlier, we have the following result.
Test of hypotheses
EXAMPLES
Confidence Interval
EXAMPLES
• Verify that the pooled standard deviation, sp = 2.7
and the test statistic value is, to = -0.35.
• At 5% level of significance, we do not reject the null
hypothesis and conclude that there is no sufficient
evidence to say the means of the two catalysts differ.
EXAMPLES
EXAMPLES
Dependent Populations
Pairwise test
• So far, we assumed the populations under
study to be independent.
• What happens when we need to compare,
say, two assembling methods using the
same set of operators?
• Obviously, the populations are not
independent.
• So, what do we do?
• Verify that the calculated value of the statistic is 6.05.
• Since this value is greater than the table value at 5% level, we
reject the null hypothesis and conclude at 5% level that there is
sufficient evidence to say that Karlsruhe method produces more
strength on the average than the Lehigh method.
Verify that 90% confidence interval for the difference in the means
is given by (-4.79, 7.21).
Inference on the variances of two
normal populations
Test on the variances
EXAMPLES
Confidence interval
s

s
f1 / 2, n2 1,n1 1 

f / 2, n2 1,n1 1
s

s
2
1
2
2
2
1
2
2
Examples
2
1
2
2
Large Sample Test for proportions
To test H0: p1 = p2 vs H1: p1  p2, the test statistic is
z
Pˆ1  Pˆ2  ( p1  p2 )
p1 (1  p1 ) p2 (1  p2 )

n1
n2
Decision Rule: Reject the null hypothesis if the absolute
value of the calculated value of the above statistic exceeds
the table value z/2
HOMEWORK PROBLEMS
Sections 5.1 through 5.7
1-5, 9, 11, 13, 14, 16, 17, 20, 22, 30-33, 4042, 46, 48-51, 53, 54