Section 9.1 ~
Fundamentals of Hypothesis Testing
Introduction to Probability and Statistics
FALL 2015
Sec. 9.1
Objective

After this section you will understand the
goal of hypothesis testing and the basic
structure of a hypothesis test, including how
to set up the null and alternative hypotheses,
how to determine the possible outcomes of a
hypothesis test, and how to decide between
these possible outcomes.
Statistical Claims
• “Of our 350 million users, more than 50% log on to
Facebook everyday”
• “Using Gender Choice could increase a woman’s chance
of giving birth to a baby girl up to 80%”
• “According to the U.S. Census Bureau, Current
Population Surveys, March 1998, 1999, and 2000, the
average salary of someone with a high school diploma
is $30,400 while the average salary of someone with
a Bachelor's Degree is $52,200.”
• How could we determine whether these claims are
true or not?
– Hypothesis Testing
Sec. 9.1
Formulating the Hypothesis

A hypothesis is a claim about a population
parameter
Could either be a claim about a population mean, μ,
or a population proportion, p
 All of the claims on the previous slide would be
considered hypotheses


A hypothesis test is a standard procedure for
testing a claim about a population parameter

There are always at least two hypotheses in any
hypothesis test; the null & alternative hypotheses
Sec. 9.1
Null Hypothesis

The null hypothesis, represented as H 0 (read
as “H-naught”), is the starting assumption for
a hypothesis test

The null hypothesis always claims a specific value
for a population parameter and for our purposes,
will take the form of an equality

Take the claim, “Using Gender Choice could increase a
woman’s chance of giving birth to a baby girl up to
80%” for example.

If the product did not work, it would be expected that there
would be an approximately equally likely chance of having
either a boy or a girl. Therefore, the null hypothesis (the
claim not working) would be:
null hypothesis -
H 0: p  0.5
Sec. 9.1
Alternative Hypothesis

The alternative hypothesis, represented as H a , is a claim that
the population parameter has a value that differs from the
value claimed in the null hypothesis, or in other words, the claim
does hold true

The alternative hypothesis can take one of the following forms:

left tailed: H a : population parameter < claimed value

Ex. ~ A manufacturing company claims that their new
hybrid model gets 62 mpg. A consumer group claims that
the mean fuel consumption of this vehicle is less than 62
mpg.
 This alternative hypothesis would be considered lefttailed since the claimed value is smaller (or to the left)
of the null value
null hypothesis − 𝐻0 : 𝜇 = 62
alternative hypothesis - H a :   62 mpg
Sec. 9.1
Alternative Hypothesis

The alternative hypothesis, represented as H a , is a claim that
the population parameter has a value that differs from the
value claimed in the null hypothesis, or in other words, the claim
does hold true

The alternative hypothesis can take one of the following forms:

right tailed: H a : population parameter > claimed value

Ex. ~ The claim that Gender Choice increases a woman’s chance of
having a baby girl up to 80% would be testing values above the null
value of .5, and would therefore be right-tailed
null hypothesis − 𝐻0 : 𝑝 = .5
alternative hypothesis - H a : p  0.5
Sec. 9.1
Alternative Hypothesis Cont’d…

two tailed H a : population parameter  claimed value

Ex. ~ A wildlife biologist working in the African
savanna claims that the actual proportion of
female zebras in the region is different from the
accepted proportion of 50%.

Since the claim does not specify whether the alternative
hypothesis is above 50% or below 50%, it would be
considered two-tailed in which case the values above and
below would be tested
null hypothesis − 𝐻0 : 𝑝 = .5
alternative hypothesis - H a : p  0.5
This will help you to get the
hang of it…
APSTATSGUY - Null &
Alternative Hypotheses
What do you think?
Let’s Practice!


With a partner (if you choose)…
Given a situation, identify the following
pieces of information:
Population Parameter
b) Null Hypothesis
c) Alternative Hypothesis
d) Left-Tailed? Right-Tailed? Two-Tailed?
a)
Sec. 9.1
Possible Outcomes of a Hypothesis Test

There are two possible outcomes to a
hypothesis test:

Reject the null hypothesis in which case we have
evidence in support of the alternative hypothesis
 Not reject the null hypothesis in which case we do
not have enough evidence to support the alternative
hypothesis

NOTE – Accepting the null hypothesis is not a possible
outcome since it is the starting assumption.


The test may provide evidence to NOT REJECT the null
hypothesis, but that does not mean that the null hypothesis
is true
Be sure to formulate the null and alternative
hypotheses prior to choosing a sample to avoid
bias
Sec. 9.1
Example 1

For the following case, describe the possible
outcomes of a hypothesis test and how we
would interpret these outcomes:

The manufacturer of a new model of hybrid car
advertises that the mean fuel consumption is
equal to 62 mpg on the highway (μ = 62 mpg). A
consumer group claims that the mean is less than
62 mpg
(μ < 62 mpg).

Possible outcomes:
a) Reject the null hypothesis of μ = 62 mpg in which case
we have evidence in support of the consumer group’s
claim that the mean mpg of the new hybrid is less
than 62
b) Do not reject the null hypothesis, in which case we
lack evidence to support the consumer group’s claim
 Note – this does not necessarily imply that the
manufacturer’s claim is true though
HOMEWORK

Pg 378-379 #2,3,7,9,12,15-22all
 When
completing #15-22all, follow these
directions:

Given a situation, identify the following pieces of
information:
a)
b)
c)
d)
e)
f)
Population Parameter
Null Hypothesis
Alternative Hypothesis
Left-Tailed? Right-Tailed? Two-Tailed?
What would it mean to reject the null?
What would it mean to not reject the null?
Sec. 9.1
Drawing a Conclusion from a Hypothesis Test

Using the claim that Gender Choice could increase a
woman’s chance of giving birth to a baby girl up to
80%, suppose that a sample produces a sample
proportion of, pˆ  0.52 .

Although this supports the alternative hypothesis of p  0.5 ,
is it enough evidence to reject the null hypothesis?


This is where statistical significance comes into play (introduced
in section 6.1)
Recall that something is considered to be statistically
significant if it most likely DID NOT occur by chance
FROM 6.1 MASTERY ASSESSMENT:

Choose the ONE & ONLY event below that depicts results that are statistically
significant.
a) In 500 tosses of a coin, you observe 257 heads.
b) In conducting a survey on eye color, a researcher claims that she randomly
selected a sample of 20 people, all of which had blue eyes.
c) A jury serving on a murder trial consists of 5 women and 7 men.
d) When rolling a fair die 60 times, you observe 20 results that are less than 3.
Sec. 9.1
Statistical Significance

There are two levels of statistical significance:



The 0.05 level ~ which means that if the probability
of a particular result occurring by chance is less than
0.05, or 5%, then it is considered to be statistically
significant at the 0.05 level
The 0.01 level ~ which means that if the probability
of a particular result occurring by chance is less than
0.01, or 1%, then it is considered to be statistically
significant at the 0.01 level
The 0.01 level would represent a stronger significance
than the 0.05 level
Sec. 9.1
Hypothesis Test Decisions
Based on Levels of Statistical Significance

We decide the outcome of a hypothesis test by
comparing the actual sample result (mean or
proportion) to the result expected if the null
hypothesis is true (using z-scores). We must choose a
significance level for the decision.



If the chance that the sample result occurred by chance is
less than 0.01, then the test is statistically significant at the
0.01 level and offers STRONG evidence for rejecting the null
hypothesis.
If the chance that the sample result occurred by chance is
less than 0.05, then the test offers MODERATE evidence for
rejecting the null hypothesis.
If the chance that the sample result occurred by chance is
greater than the chosen level of significance (0.01 or 0.05),
then we DO NOT reject the null hypothesis.
Sec. 9.1
P-Values

A P-Value, or probability value, is the value that
represents the probability of selecting a sample at
least as extreme as the observed sample





In other words, it is the value that allows us to determine if
something is statistically significant or not
NOTE ~ notice that the P-Value is represented using a capitol
P, whereas the population proportion is represented using a
lowercase p.
We will learn how to actually calculate the P-Value in the
following sections
A small P-value indicates that the observed result is
unlikely (therefore statistically significant) and
provides evidence to reject the null hypothesis
A large P-value indicates that the sample result is not
unusual, therefore not statistically significant - or
that it could easily occur by chance, which tells us to
NOT reject the null hypothesis
Sec. 9.1
Example 2

You suspect that a coin may have a bias toward landing
tails more often than heads, and decide to test this
suspicion by tossing the coin 100 times. The result is that
you get 40 heads (and 60 tails). A calculation (not shown
here) indicates that the probability of getting 40 or fewer
heads in 100 tosses with a fair coin is 0.0228. Find the Pvalue and level of statistical significance for your result.
Should you conclude that the coin is biased against heads?


The P-Value is 0.0228
This value is smaller than 5% (.05), but not smaller than 1% (.01),
so it is statistically significant at the 0.05 level which gives us
moderate reason to reject the null hypothesis and conclude that
the coin is biased against heads
Sec. 9.1
Putting It All Together
Step 1. Formulate the null and alternative hypotheses, each of which must
make a claim about a population parameter, such as a population
mean (μ) or a population proportion (p); be sure this is done
before drawing a sample or collecting data. Based on the form of
the alternative hypothesis, decide whether you will need a left-,
right-, or two-tailed hypothesis test.
Step 2. Draw a sample from the population and measure the sample
statistics, including the sample size (n) and the relevant sample
statistic, such as the sample mean (x) or sample proportion (p).
Step 3. Determine the likelihood of observing a sample statistic (mean or
proportion) at least as extreme as the one you found under the
assumption that the null hypothesis is true. The precise
probability of such an observation is the P-value (probability
value) for your sample result.
Step 4. Decide whether to reject or not reject the null hypothesis, based
on your chosen level of significance (usually 0.05 or 0.01, but
other significance levels are sometimes used).
HOMEWORK
Pg 378 – 379
#6, 11, 13, 23-28all
