Probability & Statistical Inference Lecture 6

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 Significance Level

Collect Evidence 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

Tests of Statistical Hypotheses

Decision criteria for testing H0: = 50 centimeters per second versus H1: 50 centimeters per second.

Example

Some Definitions

There is a chance you could be wrong!

Errors in Hypothesis TestsActual

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:

Upper tail test

Lower tail test

01

00

µ µ : Hµ µ : H

01

00

µ µ : Hµ µ : H

01

00

µ µ : Hµ µ : H

General Steps in Hypotheses testing1. 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:

nXZ

/

__

0

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:

nXZ

/0

__

0

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 hours. 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.

83725.0__

x

Hypothesis Tests on the Mean, σ2 unknown

Two tail test:

Upper tail test

Lower tail test

01

00

µ µ : Hµ µ : H

01

00

µ µ : Hµ µ : H

01

00

µ µ : Hµ µ : H

Example

Example The sample mean and the standard deviation

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

83725.0__

x

Normal probability plot of the coefficient of restitution data from the example.

Normal probability plot

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:

a) How many degrees of freedom are there on the t-test statistic

b) Fill in the missing quantatiesc) Test the hypotheses H0: µ=34.5 versus H1: µ≠34.5

using α=0.05

Variable N Mean StDev SE Mean 95%CI t X 16 35.274 1.783 ? (34,324,36.224) ?

Tests on a Population Proportion Large-Sample Tests on a Proportion

An appropriate test statistic is

Tests on a Population Proportion