1. Probability & Statistical Inference Lecture 6
MSc in Computing (Data Analytics)

2. Lecture Outline

3. 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.

4. 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.

5. Judicial Analogy Significance Level Hypothesis Collect Evidence
Decision Rule

6. 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

7. 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.

8. 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

9. Example Tests of Statistical Hypotheses
Decision criteria for testing H0: = 50 centimeters per second versus H1:  50 centimeters per second.

10. Some Definitions
There is a chance you could be wrong!

11. Errors in Hypothesis Tests
Actual
Decision H0 H1
Correct
Type II Error
Type I error
Sometimes the type I error probability is called the significance level, or the -error, or the size of the test

12. 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 = , so the power of this test is 1 - b = = when m = 52.

13. 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.

14. Critical Regions
Two tail test:
Upper tail test
Lower tail test

15. General Steps in Hypotheses testing
From the problem context, identify the parameter of interest.
State the null hypothesis, H0 .
Specify an appropriate alternative hypothesis, H1.
Choose a significance level, .
Determine an appropriate test statistic.
State the rejection region for the statistic.
Compute any necessary sample quantities, substitute these into the equation for the test statistic, and compute that value.
Decide whether or not H0 should be rejected and report that in the problem context.

16. Tests on the Mean of a Normal Dist, σ Known
Hypothesis Tests on the Mean
We wish to test:
The test statistic is:

17. 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

18. Example

19. Example
We can solve this problem by using the 8 steps as follows:

20. Example

21. Recap Assumptions The population variance σ is known.
The sample means are normally distributed. (Invoke the CLT)

22. 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 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.

23. Hypothesis Tests on the Mean, σ2 unknown

24. Two tail test:
Upper tail test
Lower tail test

25. Example

26. Example The sample mean and the standard deviation
and s = 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

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

28. Example

29. 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, and
Test the hypotheses H0: µ=22.5 versus H1: µ≠22.5 using α=0.05
Consider this computer output:
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
Variable N
Mean
StDev
SE Mean
95%CI t X 16
35.274
1.783 ?
(34,324,36.224)

30. Tests on a Population Proportion
Large-Sample Tests on a Proportion
An appropriate test statistic is

31. Tests on a Population Proportion