Transcript Lecture 29 - Hypothesis Testing Introduction

Testing Hypotheses
about a Population
Proportion
Lecture 29
Sections 9.1 – 9.3
Fri, Nov 12, 2004
Discovering Characteristics of a
Population


Our question about a population must first be
described in terms of a population parameter.
Then our question about that parameter
generally falls into one of two categories.

What is the value of the parameter?


That is, estimate its value.
Does the evidence tend to support or refute a claim
about the value of the parameter?

That is, test a hypothesis concerning the parameter.
Example




A standard assumption is that a newborn baby is
as likely to be a boy as it is to be a girl.
However, some believe that boys are more likely.
A random sample of 1000 live births shows that
520 are boys and 480 are girls.
Does this evidence support or refute the
standard assumption?
The Steps of Testing a
Hypothesis

The basic steps are
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.


See page 519.
Step 1: State the Null and
Alternative Hypotheses


Let p = proportion of live births that are boys.
The null and alternative hypotheses are
H0: p = 0.50.
 H1: p > 0.50.

State the Null and Alternative
Hypotheses

The null hypothesis should state a hypothetical
value p0 for the population proportion.


H0: p = p0.
The alternative hypothesis must contradict the
null hypothesis in one of three ways:
H1: p < p0. (Direction of extreme is left.)
 H1: p  p0. (Direction of extreme is left and right.)
 H1: p > p0. (Direction of extreme is right.)

Explaining the Data


The observation is 520 males out of 1000 births,
or 52%.
Since we did not observe 50%, how do we
explain the discrepancy?
Chance, or
 The true proportion is not 50%, but something
larger, maybe 52%.

Step 2: State the Significance Level



The significance level  should be given in the
problem.
If it isn’t, then use  = 0.05.
In this example, we will use  = 0.05.
The Sampling Distribution of p^



To decide whether the sample evidence is
significant, we will compare the p-value to .
From the value of , we may find the critical
value(s).
 is the probability that the sample data are at
least as extreme as the critical value(s), if the null
hypothesis is true.
The Sampling Distribution of p^

Therefore, when we compute the p-value, we do
it under the assumption that H0 is true, i.e., that
p = p0.
The Sampling Distribution of p^

We know that the sampling distribution of p^ is
normal with mean p and standard deviation
 pˆ 

p1  p 
n
Thus, we assume that p^ has mean p0 and
standard deviation:
 pˆ 
p0 1 p0 
n
Step 3: The Test Statistic


Test statistic – The z-score of p^, under the
assumption that H0 is true.
Thus,
Z
pˆ   pˆ
 pˆ

pˆ  p0
p0 1  p0 
n
The Test Statistic

In our example, we compute
(.50)1  .50
 pˆ 
 0.0158.
1000

Therefore, the test statistic is
pˆ  0.50
Z
0.0158

Now, to find the value of the test statistic, all we
need to do is to collect the sample data and
substitute the value of p^.
Computing the Test Statistic


In the sample, p^ = 0.52.
Thus,

z = (0.52 – 0.50)/0.0158 = 1.26.
Step 4: Compute the p-value



To compute the p-value, we must first check
whether it is a one-tailed or a two-tailed test.
We will compute the probability that Z would be
at least as extreme as the value of our test
statistic.
If the test is two-tailed, then we must take into
account both tails of the distribution to get the pvalue.
Compute the p-value


In this example, the test is one-tailed, with the
direction of extreme to the right.
So we compute
P(Z > 1.26) = 0.1038.
Compute the p-value


An alternative is to evaluate
normalcdf(0.52, E99, 0.50, 0.0158)
on the TI-83.
It should give the same answer (except for
round-off).
Step 5: State the Conclusion


Since the p-value is greater than , we should not
reject the null hypothesis.
State the conclusion in a sentence.

“The data do not support the claim, at the 5% level
of significance, that more than 50% of live births
are male.”
Testing Hypotheses on the TI-83





The TI-83 has special functions designed for
hypothesis testing.
Press STAT.
Select the TESTS menu.
Select 1-PropZTest…
Press ENTER.

A window with several items appears.
Testing Hypotheses on the TI-83






Enter the value of p0. Press ENTER and the down
arrow.
Enter the numerator x of p^. Press ENTER and the
down arrow.
Enter the sample size n. Press ENTER and the down
arrow.
Select the type of alternative hypothesis. Press the
down arrow.
Select Calculate. Press ENTER.
(You may select Draw to see a picture.)
Testing Hypotheses on the TI-83

The display shows
The title “1-PropZTest”
 The alternative hypothesis.
 The value of the test statistic Z.
 The p-value.
 The value of p^.
 The sample size.


We are interested in the p-value.