Aim: How do we test hypotheses that compare means of two groups?

HW: complete last two questions on homework slides

Comparing the Means of Two Groups

• Example of such a senior– Does the mean age of nursing students who enroll

at a community college differ from the mean age of nursing students who enroll at a university?• Here, the hypotheses are

0 1 2

1 1 2

:

:

H

H

1

2

Mean age of those in community colleges

Mean age of those at university

Comparing the Means of Two Groups

• Two ways to state hypotheses

0 1 2

1 1 2

:

:

H

H

0 1 2

1 1 2

: 0

: 0

H

H

If there is no difference in population means, subtracting them will give a difference of zero!If they are different, subtracting will give a number other than zero.

Assumption for the Test to Determine the Difference Between Two Means

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

2. The populations from which the samples were obtained must be normally distributed, and the standard deviations of the variable must be known, or the sample size must be greater than or equal to 30.

Theory Behind Testing…

• The difference between two means is based on selecting pairs of samples and comparing the means of the pairs

• The population means need not be known

Other Types of Hypotheses

• Right Tailed • Left Tailed

0 1 2

1 1 2

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:

H

H

0 1 2

1 1 2

:

:

H

H

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

1 2 1 2

2 21 2

1 2

X Xz

n n

Procedure

1. State the hypotheses and identify the claim2. Find the critical value(s)3. Compute the test value4. Make the decision 5. Summarize the results

Example

• A survey found that the average hotel room rate in New Orleans is $88.42 and the average room rate in Phoenix is $80.61. Assume the data were obtained from two samples of 50 hotels each and the standard deviations were $5.62 and $4.83, respectively. At α = 0.05, can it be concluded that there is significant difference in the rates?

Solution

1. State Hypothesis and identify claim

2. Find critical value

3. Calculate test value

4. Make a decision

5. Summarize the results

Test value > critical value (reject null)

There is enough evidence to support the claim that the means are not equal to each other

0 1 2

1 1 2

:

: ( )

H

H claim

.05

1.96

1 2 1 2

2 2 2 21 2

1 2

88.42 80.61 07.45

5.62 4.8350 50

X Xz

n n

Formula for Confidence Interval

2 2 2 21 2 1 2

1 2 1 2 1 21 2 1 22 2

X X z X X zn n n n

Class Work1. Explain the difference between testing a single mean

and testing the difference between two means.2. A study conducted to see if there was a difference

between spouses and significant others in coping skills when living with or caring for a person with multiple sclerosis. These skills were measured by questionnaire responses. The results of the two groups are given on one factor, ambivalence. At α = .10, is there a difference in the means of the two groups?

v 1

1

1

2.0

.6

120

spouse

X

s

n

2

2

2

1.7

.7

34

Signficiant

X

s

n

Class Work

3. At age 9 the average weight (21.3kg) and the average height (124.5 cm) for both boys and girls are exactly the same. A random sample of 9-year-olds yielded these results. Estimat ethe mean difference in height between boys and girls with 95% confidence. Does your interval suport the given claim?

HomeworkQuestion 1

HomeworkQuestion 2