Transcript Hypothesis Tests on the Mean

Probability & Statistical Inference
Lecture 6
MSc in Computing (Data Analytics)
Lecture Outline
Hypothesis Testing

Statistical hypothesis testing and confidence interval
estimation of parameters are the fundamental
methods used at the data analysis stage of a
comparative experiment, in which the
experimenter is interested, for example, in comparing
the mean of a population to a specified value.
Example

For example, suppose that we are interested in the
burning rate of a solid propellant used to power
aircrew escape systems.

Now burning rate is a random variable that can be
described by a probability distribution.

Suppose that our interest focuses on the mean
burning rate (a parameter of this distribution).

Specifically, we are interested in deciding whether or
not the mean burning rate is 50 centimeters per
second.
Judicial Analogy
Hypothesis
Collect Evidence
Significance Level
Decision Rule
Judicial Analogy



A defendant is put on trial. They are suspected of being
guilty of crime.
Determine the null hypothesis H0 and the
alternative hypothesis H1.
The null hypothesis is what you assume to be true when
you start your analysis. It is the logical opposite of what
you are tying to prove. In the judicial analogy:
H0: The defendant is innocent
H1:The defendant is guilty
Judicial Analogy



You select a significance level. In the judicial example it
is the amount of evidence needed to convict. In a court of
law there must be enough evidence to convict ‘beyond a
reasonable doubt’.
You collect evidence.
You use the decision rule to make a judgement. If the
evidence is


sufficiently strong, reject the null hypothesis. The defendant is
proven guilty
not strong enough, do not reject the null hypothesis.
Coin Example

You suspect that a coin is not fair and set out to prove that it
is not fair
H0: The coin is fair
H1: The coin is not fair



Significance level: If you observe more than 8 head or tails
coin tosses out of ten you conclude the coin is not fair
otherwise you state that there is not enough evidence
Toss the coin ten times and count the number of heads and
tails
You evaluate the data using your decision rule that there is


Enough evidence to reject the assumption that the coin is fair
Not enough evidence to reject the assumption that the coin is fair
Example
Tests of Statistical Hypotheses
Decision criteria for testing H0: = 50 centimeters per second versus H1: 
50 centimeters per second.
Some Definitions
There is a chance you could be wrong!
Errors in Hypothesis Tests
Actual
Decision
H0
H1
H0
Correct
Type II Error
H1
Type I error
Correct
Sometimes the type I error probability is called
the significance level, or the -error, or the size of
the test
Errors in Hypothesis Tests

β = P(type II error) = P(fail to reject H0 when H0 is
false)

The power is computed as 1 - β, and power can be
interpreted as the probability of correctly rejecting a
false null hypothesis. We often compare statistical
tests by comparing their power properties.

For example, consider the propellant burning rate
problem when we are testing H 0 : m = 50 centimeters
per second against H 1 : m not equal 50 centimeters
per second . Suppose that the true value of the mean
is m = 52. When n = 10, we found that b = 0.2643, so
the power of this test is 1 - b = 1 - 0.2643 = 0.7357
when m = 52.
Which Hypothesis is of interest

Suppose you have a question about the quantity of
cereal is a box of cornflakes. You can use one of
three types of test:
 A two tail test if you suspect the true mean is
different rather than claimed.
 An upper-tail test if you suspect the true mean is
higher than claimed
 A lower-tailed test if you suspect that that the
true mean is lower than claimed.
Critical Regions

Two tail test:
H 0 : µ  µ0
H1 : µ  µ0

Upper tail test
H 0 : µ  µ0
H1 : µ  µ0

Lower tail test
H 0 : µ  µ0
H1 : µ  µ0
General Steps in Hypotheses testing
1.
From the problem context, identify the parameter of
interest.
2.
State the null hypothesis, H0 .
3.
Specify an appropriate alternative hypothesis, H1.
4.
Choose a significance level, .
5.
Determine an appropriate test statistic.
6.
State the rejection region for the statistic.
7.
Compute any necessary sample quantities, substitute these
into the equation for the test statistic, and compute that
value.
8.
Decide whether or not H0 should be rejected and report
that in the problem context.
Tests on the Mean of a Normal Dist, σ Known

Hypothesis Tests on the Mean
 We wish to test:

The test statistic is:
__
X
Z0 
/ n
Tests on the Mean of a Normal Dist, σ Known
 Reject H0 if the observed value of the test statistic
z0 is either:
z0 > z/2 or z0 < -z/2

Fail to reject H0 if
-z/2 < z0 < z/2
Example
Example

We can solve this problem by using the 8 steps as follows:
__
X  0
Z0 
/ n
Example
Recap
Assumptions
• The population variance σ is known.
•The sample means are normally distributed. (Invoke the CLT)
Exercises

The life in hours of a battery is known to be
approximately normally distributed with a standard
deviation σ=1.25 hours.
A random sample of 40 batteries
__
has a mean life of x  0.83725hours.


Is there evidence to support that battery life exceeds 40.5
hours? Use α=0.05.
The mean water temperature downstream from a power
plant cooling tower discharge pipe should be no more than
38oC. Past experience has indicated the standard deviation
of the temperature is 1.1o. The water temperature
measured on 35 randomly chosen days and the average
temperature is found to be 37oC.

Is there evidence that the water temperature is acceptable at
α=0.05.
Hypothesis Tests on the Mean, σ2 unknown

Two tail test:
H 0 : µ  µ0
H1 : µ  µ0

Upper tail test
H 0 : µ  µ0
H1 : µ  µ0

Lower tail test
H 0 : µ  µ0
H1 : µ  µ0
Example
Example

The sample mean and the standard deviation
x  0.83725
and s = 0.02456. The normal probability plot of the data on the
next slides supports the assumption that the sample means
come from a normal distribution. Use the 8 steps to test that
the mean coefficient of restitution exceeds 0.82
__
Normal probability plot
Normal probability plot
of the coefficient of
restitution data from
the example.
Example
Exercise

An article in a journal describes a study of thermal inertia
properties of autoclaved aerated concrete used as building
material. Five samples of the material was tested in a structure,
and the average interior temperate (oC) reported were as
follows: 23.01, 22.22, 22.04, 22.62 and 22.59.


Test the hypotheses H0: µ=22.5 versus H1: µ≠22.5 using α=0.05
Consider this computer output:
Variable N
X 16
a)
b)
c)
Mean
35.274
StDev
1.783
SE Mean
?
95%CI
t
(34,324,36.224)
?
How many degrees of freedom are there on the t-test
statistic
Fill in the missing quantaties
Test the hypotheses H0: µ=34.5 versus H1: µ≠34.5 using
α=0.05
Tests on a Population Proportion

Large-Sample Tests on a Proportion
An appropriate test statistic is
Tests on a Population Proportion