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Mathematics & Engineering Courses

MTHE 224

Prof.

Fall

Description

Week 5: Conﬁdence Intervals and
Hypothesis Testing
Goals:
• Study conﬁdence intervals based on a normal population distribution
• Study large-sample conﬁdence intervals
• Study Hypothesis Testing
Suggested Textbook Readings: Chapter 7: 7.1 - 7.2, Chapter 8: 8.1, 8.2
Practice Problems:
Section 7.1: 1, 3, 5, 7
Section 7.2: 13, 15, 17
Section 8.1: 7, 9, 11
Section 8.2: 1 Week 5: Conﬁdence Intervals and Hypothesis Testing 2
What is a Conﬁdence Interval?
It is important not to think of a conﬁdence interval as the probability the true mean lies
inside X ± 1.96 √n is 0.95. The correct interpretation is more elusive.
Conﬁdence Interval Meaning: the long run proportion of random samples of size n
from a normal (µ,σ) population that have true mean µ inside the calculated conﬁdence
interval is 0.95.
Deﬁnition: The expression √σ in the CI expression is called the standard error (syn-
onomous with the standard deviation of the sample mean of a random sample of size
n)
We can consider other level of conﬁdence. Similar to 95% CI, other level of conﬁdence,
for example 100(1 − α) CI, can be achieved by replacing 1.96 with the appropriate
standard normal critical value α/2, that is
▯ ▯
σ σ
¯ − zα/2 ·√ n,¯ + zα/2 ·√ n
Example 1: Calculate a 90% conﬁdence interval for the true mean of a normal popu-
lation with parameters σ =6 .0, n = 34, and ¯ = 80.
Choice of Sample Size
Example 2: (Example 7.4, page 259) Extensive monitoring of a computer time-sharing
system has suggested that response time to a particular editing command is normally dis-
tributed with standard deviation 25 millisec. A new operating system has been installed,
and we wish to estimate the true average response time µ for the new environment. As-
suming that response times are still normally distributed with σ = 25, what sample size
is nessary to ensure that the resulting 95% conﬁdence interval has a width of (at most)
10?
In general, the sample size required to get an interval width w satisﬁes the condition
that σ
w =2 · z α/2√
n
MTHE 224 Fall 2012 Week 5: Conﬁdence Intervals and Hypothesis Testing 3
Large Sample Conﬁdence Intervals
To generalize this theory:
1. Because of the Central Limit theorem, if the sample size n is large enough, we
know that the sample mean X (when calculated from a random sample) has an
approximate normal distribution.
2
2. When n is large, we also expect S to approximate V [X] very well. (Where X has
the distribution of the random sample.)
Proposition: If n is suﬃciently large (n> 40), the rv
¯
Z = X −√µ
S/ n
has approximately a standard normal distribution, which implies that
s
¯ ± zα/2 ·√ n
is a 100(1 − α) conﬁdence interval for µ.
Example 3: (Example 7.6 and 7.7, page 264 and 265) The sample observations
on the alternating current (AC) breakdown voltage of an insulating liquid are given.
Summary quantities include n = 48, ▯ x = 2626, and ▯ x = 144,950.
i i
(a) Find the sample mean and the sample standard deviation.
(b) Find the 95% conﬁdence interval for µ.
(c) Suppose the investigator believes that virtually all values in the population are between
40 and 70. Estimate the sample size for a 95% CI with width 2.
MTHE 224 Fall 2012 Week 5: Conﬁdence Intervals and Hypothesis Testing 4
We may creat a normal plot of the data to check normality.
X=xlsread(’example7-6.xlsx’);
normplot(X);
One-Sided Conﬁdence Intervals (Conﬁdence Bounds)
A one-sided conﬁdence bound results from replacing z α/2by z α In all cases the conﬁ-
dence level is approximately 100(1 − α)%.
A large-sample upper conﬁdence bound for µ is
µ< x¯ + zα· √s
n
and a large-sample lower conﬁdence bound for µ is
µ> x¯ − z · √s
α n
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