Download Making Decisions about a Population Mean with Confidence

Making Decisions about
a Population Mean with
Confidence
Lecture 35
Sections 10.1 – 10.2
Fri, Mar 25, 2005
Introduction

In Chapter 10 we will ask the same basic
questions as in Chapter 9, except they will
concern the mean.
Find an estimate for the mean.
 Test a hypothesis about the mean.

The Steps of Testing a
Hypothesis (p-Value Approach)
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1. State the null and alternative hypotheses.
2. State the significance level.
3. Compute the value of the test statistic.
4. Compute the p-value.
5. State the conclusion.
The Hypotheses

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The null and altenative hypotheses will be
statements concerning .
Null hypothesis.

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H0:  = 0.
Alternative hypothesis (choose one).
H1:   0.
 H1:  < 0.
 H1:  > 0.

Level of Significance
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The level of significance is the same as before.
If the value is not given, assume that  = 0.05.
The Test Statistic

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The choice of test statistic will depend on the
sample size and what is known about the
population.
For the moment, we will assume that  is known
for the population.
Later we will consider the case where  is
unknown.
The Sampling Distribution ofx

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If the population is normal, then the distribution
ofx is also normal, with mean 0 and standard
deviation /n.
That is,x is N(0, /n).
Note that this assumes that  is known.
See p. 500, the Central Limit Theorem.
The Sampling Distribution ofx

Therefore, the test statistic is
x  0
Z
/ n

It is exactly standard normal.
The Sampling Distribution ofx

On the other hand, if
the population is not normal,
 but the sample size is at least 30,

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then the distribution ofx is approximately normal,
with mean 0 and standard deviation /n.
That is,x is approximately N(0, /n).
Note that we are still assuming that  is known.
See p. 500, the Central Limit Theorem.
The Sampling Distribution ofx

Therefore, the test statistic is
x  0
Z
/ n


It is approximately standard normal.
The approximation is good enough that we can
use the normal tables.
Decision Tree
Is  known?
yes
no
Decision Tree
Is  known?
yes
Is the population
normal?
yes
no
no
Decision Tree
Is  known?
yes
Is the population
normal?
yes
no
Z
X 
/ n
no
Decision Tree
Is  known?
yes
no
Is the population
normal?
yes
no
X 
Z
/ n
Is n  30?
yes
no
Decision Tree
Is  known?
yes
no
Is the population
normal?
yes
no
Is n  30?
X 
Z
/ n
yes
Z
X 
/ n
no
Decision Tree
Is  known?
yes
no
Is the population
normal?
yes
no
Is n  30?
X 
Z
/ n
yes
Z
X 
/ n
no
Give
up
Decision Tree
Is  known?
yes
no
Is the population
normal?
yes
no
TBA
Is n  30?
X 
Z
/ n
yes
Z
X 
/ n
no
Give
up
Example

See Example 10.1, p. 569 – Too Much Carbon
Monoxide? ( known).
Let’s Do It!

Let’s Do It! 10.2, p. 573 – Completing a Maze.
Hypothesis Testing on the TI-83
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Press STAT.
Select TESTS.
Select Z-Test. Press ENTER.
A window appears requesting information.
Select Data if you have the sample data entered
into a list.
Otherwise, select Stats.
Hypothesis Testing on the TI-83
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Assuming you selected Stats,
Enter 0, the hypothetical mean.
Enter . (Remember,  is known.)
Enterx.
Enter n, the sample size.
Select the type of alternative hypothesis.
Select Calculate and press ENTER.
Hypothesis Testing on the TI-83

A window appears with the following
information.
The title “Z-Test.”
 The alternative hypothesis.
 The value of the test statistic Z.
 The p-value of the test.
 The sample mean.
 The sample size.

Example


Re-do Example 10.1 on the TI-83 (using Stats).
The TI-83 reports that
z = –2.575.
 p-value = 0.005012.

Hypothesis Testing on the TI-83

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Suppose we had selected Data instead of Stats.
Then somewhat different information is
requested.
Enter the hypothetical mean.
 Enter .
 Identify the list that contains the data.
 Skip Freq (it should be 1).
 Select the alternative hypothesis.
 Select Calculate and press ENTER.

Hypothesis Testing on the TI-83

Why enter  if the TI-83 is capable of
computing the standard deviation from the data?
Example



Re-do Example 10.1 on the TI-83 (using Data).
Enter the data in the chart on page 570 into list
L1.
The TI-83 reports that
z = -2.575.
 p-value = 0.005012.
 x = 12.528.
 s = 4.740 ( 4.8).

Let’s Do It!

Re-do Let’s Do It! 10.2, p. 573, using the TI-83.