CH9: Testing the Difference Between Two Means, Two ...

[Pages:67]CH9: Testing the Difference Between Two Means, Two Proportions, and Two Variances

CH9: Testing the Difference Between Two Means or Two Proportions

Santorico - Page 343

Section 9-1 Testing the Difference Between Two Means: Using the Z Test

Suppose we are interested in determining if a certain medication relieves patients' headaches.

We give the drug/treatment to one group and give a placebo to a control group and compare the mean incidences of patient relief from the headache between the two groups.

If the treatment group had a statistically significant improvement in headache symptoms over the control group, then we can conclude the drug works.

CH9: Testing the Difference Between Two Means or Two Proportions

Santorico - Page 344

So our question might be, "Is the mean incidence of headache relief different for the two groups?"

Let 1 mean headache relief from treatment group and 2 mean headache relief from control group.

Then our hypotheses would be:

H0 : H1 :

Alternatively, we could state the hypotheses as: H0 : H1 :

CH9: Testing the Difference Between Two Means or Two Proportions

Santorico - Page 345

Assumptions for the Test to Determine the Difference Between Two Means

The samples must be independent of each other. That is, there can be no relationship between the subjects in each sample.

The populations from which the samples come must be (approximately) normally distributed or the sample sizes of both groups should be at least 30.

The standard deviations of both populations must be known.

CH9: Testing the Difference Between Two Means or Two Proportions

Santorico - Page 346

We can compare the groups by the difference in their

population means, 1 2, where 1 is the population mean for group 1 and 2 is the population mean for group 2.

We estimate 1 2 withx 1 x2

Thestandard deviation of x1 x2 is

2 1

2 2

n1 n2

When both populations are normally distributed or the

samples size foreach groupis at least 30, then x1 x2 has a

normal distribution.

CH9: Testing the Difference Between Two Means or Two Proportions

Santorico - Page 347

Formula for the z test for Comparing Two Means from Independent Populations

H0 :1 2 k (or k or k)

Note: We often k 0, but it doesn't have to be.

Test value:

z*

(x1

x2 ) (1

2 1

2 2

2

)

(x1 x2 ) k

2 1

2 2

n1 n2

n1 n2

CH9: Testing the Difference Between Two Means or Two Proportions

Santorico - Page 348

The observed difference between the sample means may be due to chance, in which case the null hypothesis will not be rejected.

If the difference is statistically significant, the null hypothesis is rejected and the researcher can conclude the population means are different.

The same approach to finding critical values and P-values that was used in Section 8-2 will be used here (Table E or Table F with d.f. = ).

CH9: Testing the Difference Between Two Means or Two Proportions

Santorico - Page 349

Example: Dr. Cribari would like to determine if there is a statistically significant difference between her two Math 2830 classes. To make this comparison she will compare the results from exam 1. Class one had 35 students take the exam with a mean of 82.6 and a population standard deviation of 1.41. Class two had 32 students take the exam with a mean of 84 and a population standard deviation of 3.63. Can Dr. Cribari conclude that there is difference in the mean test grades between the two classes at =0.05? Ho: ? 1 = ? 2 Ho: ? 1 ? 2

Step 1 State the hypotheses and identify the claim.

H0 : 1 2 H1 : 1 2 CLAIM

CH9: Testing the Difference Between Two Means or Two Proportions

Santorico - Page 350

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