Download Section 2

Lesson 10 - 2
Testing Claims about a Population
Mean Assuming the Population
Standard Deviation is Known
Objectives
• Understand the logic of hypothesis testing
• Test a claim about a population mean with σ known
using the classical approach
• Test a claim about a population mean with σ known
using P-values
• Test a claim about a population mean with σ known
using confidence intervals
• Understand the difference between statistical
significance and practical significance
Vocabulary
• Statistically Significant – when observed results are
unlikely under the assumption that the null hypothesis
is true. When results are found to be statistically
significant, we reject the null hypothesis
• Practical Significance – refers to things that are
statistically significant, but the actual difference is not
large enough to cause concern or be considered
important
Hypothesis Testing Approaches
• Classical
– Logic: If the sample mean is too many standard deviations
from the mean stated in the null hypothesis, then we reject the
null hypothesis (accept the alternative)
• P-Value
– Logic: Assuming H0 is true, if the probability of getting a
sample mean as extreme or more extreme than the one
obtained is small, then we reject the null hypothesis (accept
the alternative).
• Confidence Intervals
– Logic: If the sample mean lies in the confidence interval about
the status quo, then we fail to reject the null hypothesis
Classical Approach
-zα/2
-zα
zα/2
zα
Critical Regions
Test Statistic:
x – μ0
z0 = ------------σ/√n
Reject null hypothesis, if
Left-Tailed
Two-Tailed
Right-Tailed
z0 < - zα
z0 < - zα/2
or
z0 > z α/2
z 0 > zα
P-Value Approach
z0
-|z0|
|z0|
P-Value is the
area highlighted
Test Statistic:
x – μ0
z0 = ------------σ/√n
Reject null hypothesis, if
P-Value < α
z0
P-Value Examples
For each α and observed significance level (p-value)
pair, indicate whether the null hypothesis would be
rejected.
a) α = . 05, p = .10
α < P  fail to reject Ho
b) α = .10, p = .05
P < α  reject Ho
c) α = .01 , p = .001
P < α  reject Ho
d) α = .025 , p = .05
α < P  fail to reject Ho
e) α = .10, p = .45
α < P  fail to reject Ho
Confidence Interval Approach
Confidence Interval:
x – zα/2 · σ/√n
Lower
Bound
x + zα/2 · σ/√n
Upper
Bound
μ0
Reject null hypothesis, if
μ0 is not in the confidence interval
Example 1
A simple random sample of 12 cell phone bills finds xbar = $65.014. The mean in 2004 was $50.64. Assume
σ = $18.49. Test if the average bill is different today at
the α = 0.05 level. Use each approach.
Example 1: Classical Approach
A simple random sample of 12 cell phone bills finds x-bar = $65.014. The
mean in 2004 was $50.64. Assume σ = $18.49. Test if the average bill is
different today at the α = 0.05 level. Use the classical approach.
not equal  two-tailed
X-bar – μ
65.014 – 50.64
14.374
Z0 = --------------- = ---------------------- = ------------- = 2.69
σ / √n
18.49/√12
5.3376
Zc = 1.96
Using alpha, α = 0.05 the shaded region are the
rejection regions. The sample mean would be too
many standard deviations away from the population
mean. Since z0 lies in the rejection region, we would
reject H0.
Zc (α/2 = 0.025) = 1.96
Example 1: P-Value
A simple random sample of 12 cell phone bills finds x-bar = $65.014. The
mean in 2004 was $50.64. Assume σ = $18.49. Test if the average bill is
different today at the α = 0.05 level. Use the P-value approach.
not equal  two-tailed
X-bar – μ
65.014 – 50.64
14.374
Z0 = --------------- = ---------------------- = ------------- = 2.69
σ / √n
18.49/√12
5.3376
-Z0 = -2.69
The shaded region is the probability of obtaining a
sample mean that is greater than $65.014; which is
equal to 2(0.0036) = 0.0072. Using alpha, α = 0.05,
we would reject H0 because the p-value is less than α.
P( z < Z0 = -2.69) = 0.0036
Using Your Calculator: Z-Test
• For classical or p-value approaches
• Press STAT
– Tab over to TESTS
– Select Z-Test and ENTER
•
•
•
•
Highlight Stats
Entry μ0, σ, x-bar, and n from summary stats
Highlight test type (two-sided, left, or right)
Highlight Calculate and ENTER
• Read z-critical and/or p-value off screen
From previous problem:
z0 = 2.693 and p-value = 0.0071
Example 1: Confidence Interval
A simple random sample of 12 cell phone bills finds x-bar = $65.014. The
mean in 2004 was $50.64. Assume σ = $18.49. Test if the average bill is
different today at the α = 0.05 level. Use confidence intervals.
Confidence Interval = Point Estimate ± Margin of Error
= μ ± Zα/2 σ / √n
= 50.64 ± 1.96 (18.49) / √12
Zc (α/2) = 1.96
= 50.64 ± 10.4617
65.014
μ
40.18
61.10
The shaded region is the region outside the 1- α, or 95%
confidence interval. Since the sample mean lies outside
the confidence interval, then we would reject H0.
Using Your Calculator: Z-Test
• Press STAT
– Tab over to TESTS
– Select Z-Interval and ENTER
•
•
•
•
Highlight Stats
Entry σ, x-bar, and n from summary stats
Entry your confidence level (1- α)
Highlight Calculate and ENTER
• Read confidence interval off of screen
– If μ0 is in the interval, then FTR
– If μ0 is outside the interval, then REJ
From previous problem:
u0 = 50.64 and interval (54.552, 75.476)
Therefore Reject
Summary and Homework
• Summary
– A hypothesis test of means compares whether the
true mean is either
• Equal to, or not equal to, μ0
• Equal to, or less than, μ0
• Equal to, or more than, μ0
– There are three equivalent methods of performing
the hypothesis test
• The classical approach
• The P-value approach
• The confidence interval approach
• Homework
– pg 526 – 530: 1, 3, 4, 10, 12, 17, 28, 29, 30