Probability & Statistical Inference Lecture 6

MSc in Computing (Data Analytics)

Lecture Outline Quick Recap Solutions from last week Testing the difference between two sample

means Practical Hypothesis Testing

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.

Exercise – From Last Week 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 quantitiesc) 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) ?

Testing the difference between two sample means

• It is applied to compare whether the average difference between two groups is really significant or if it is due instead to random chance.

Type of questions that can be answered with Two sample hypothesis tests. A manufacturing plant want to compare the

defective rate of items coming off two different process lines.

Whether the test results of patients who received a drug are better than test results of those who received a placebo.

The question being answered is whether there is a significant (or only random) difference in the average cycle time to deliver a pizza from Pizza Company A vs. Pizza Company B.

Difference in Means of Two Normal Distributions, Variances Known

Test Assumptions

Example

Example

Example

Confidence Interval on a Difference in Means, Variances Known

Example

Example

Difference in Means of Two Normal Distributions, Variances unknown

We wish to test:

The pooled estimator of 2:

Difference in Means of Two Normal Distributions, Variances unknown

Example

Example

Example

Confidence Interval on the Difference in Means, Variance Unknown

Example

Example

Example

Practical Hypothesis 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. Calculate the P-value using a software package of choice.

6. Decide whether or not H0 should be rejected and report that in the problem context. Reject H0 when P-Value is less than .

(Golden rule: Reject H0 for small )

Practical Examples