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Chapter 9

- Hypothesis Testing
- Two Sample Test for Means and Proportions

Introduction

- The two sample test is similar to the one sample

test, except that we are now testing for

differences between two populations rather than a

sample and a population. There are three types of

two sample tests - Hypothesis Testing with Sample Means (Large

Samples) - Hypothesis Testing with Sample Means (Small

Samples) - Hypothesis Testing with Sample Proportions (Large

Samples)

The Question to be Answered

- Is the difference between sample statistics

large enough to conclude that the populations

represented by the samples are significantly

different?

Null Hypothesis

- The H0 is that the populations are the same.
- H0 µ1 µ2
- If the difference between the sample statistics

is large enough, or, if a difference of this size

is unlikely, assuming that the H0 is true, we

will reject the H0 and conclude there is a

difference between the populations.

Null Hypothesis (cont.)

- The H0 is a statement of no difference
- The 0.05 level will continue to be our indicator

of a significant difference - We change the sample statistics to a Z score,

place the Z score on the sampling distribution

and use Appendix A to determine the probability

of getting a difference that large if the H0 is

true.

Alternate Hypothesis

- The alternate hypothesis is the research

hypothesis. - If the null hypothesis is rejected, then we will

have found evidence to support the research

hypothesis. - H1 µ1 ? µ2

Formula for Hypothesis Testing with Sample Means

(Large Samples)

Explanation of formula

- The numerator is the

difference in sample means. - The denominator is the pooled

estimate of the standard error for both samples. - The pooled estimate is calculated by using the

sample information in the following formula

The Five Step Model

- Make assumptions and meet test requirements.
- State the H0 and H1.
- Select the Sampling Distribution and Determine

the Critical Region. - Calculate the test statistic.
- Make a Decision and Interpret Results.

Example Hypothesis Testing in the Two Sample Case

- Text P. 244 Problem 9.5 b (Email messages)
- Middle class families average 8.7 email messages

and working class families average 5.7 messages. - The middle class families seem to use email more

but is the difference significant?

Problem Information

- E-Mail Messages
- Sample 1 (M.Class) Sample 2 (W.Class)
- 8.7

5.7 - S1 0.3 S2 1.1
- N1 89 N2 55

Step 1 Make Assumptions and Meet Test Requirements

- We have
- Independent Random Samples
- Level of Measurement is Interval Ratio
- Sampling Distribution is normal in shape because

we have a large sample - N1 N2 100 (in this case, N1 N2 144)

Step 2 State the Null Hypothesis

- H0 µ1 µ2
- The Null asserts there is no significant

difference between the populations. - H1 µ1? µ2
- The research hypothesis contradicts the H0 and

asserts there is a significant difference between

the populations.

Step 3 Select the Sampling Distribution and

Establish the Critical Region

- Sampling Distribution Z distribution
- Alpha (a) 0.05
- Z (critical) 1.96

Using the formula

- Compute the pooled estimate (S.E.)
- Solve for Z

Step 5 Make a Decision

- The obtained test statistic (Z 19.74) falls in

the Critical Region so reject the null

hypothesis. - The difference between the sample means is so

large that we can conclude (at a 0.05) that a

difference exists between the populations

represented by the samples. - The difference between the email usage of middle

class and working class families is significant

(Z19.74, a.05)

Two-tailed Hypothesis Test

- When a .05, then .025 of the area is

distributed on either side of the curve in area

(C ) - The .95 in the middle section represents no

significant difference between the two

populations. - The cut-off between the middle section and /-

.025 is represented by a Z-value of /- 1.96.

Factors in Making a Decision

- The use of one- vs. two-tailed tests (we are more

likely to reject with a one-tailed test) - The size of the sample (N). The larger the sample

the more likely we are to reject the H0.

Significance Vs. Importance

- As long as we work with random samples, we must

conduct a test of significance. - Significance is not the same thing as

importance. - Differences that are otherwise trivial or

uninteresting may be significant.

Significance Vs. Importance

- When working with large samples, even small

differences may be significant. - The value of the test statistic (step 4) is an

inverse function of N. - The larger the N, the greater the value of the

test statistic, the more likely it will fall in

the critical region (region of rejection) and be

declared significant.

Significance Vs Importance

- Significance and importance are different things.
- A sample outcome could be
- significant and important
- significant but unimportant
- not significant but important
- not significant and unimportant

Using SPSS to do Independent Samples Test for

Difference in Two Means

- SPSS uses a t-test rather than a z-test for both

large and small samples. - Follow guidelines in text at the end of the

chapter. - In interpreting your printout, look at the

Levenes test (shown in the first two columns F

and sig.) first. - If the p-value (sig) is greater than alpha.05,

focus on interpreting the top row of the t-test

for Equality of Means. If it is less than .05,

use the bottom row of the t-test. - If the significance level (Sig. 2-tailed) is less

than a.05, then the difference between the

sample means is significant. Report t, df, and

your a-level in your interpretation.

Formula for Hypothesis Testing with Sample

Proportions (Large Samples)

- Formula for proportions
- See next slide for how to calculate the standard

deviation of the sampling distribution and the

pooled estimate of the population proportion. - Note that you need to calculate both these

values in order to solve the denominator of the

above equation!

Calculating Pu (the Pooled Estimate of the

Population Proportion) and the Standard Deviation

of the Sampling Distribution

- To calculate Pu (the pooled estimate, see p.

231) - Standard Deviation of the S.D. (see p. 231)

Example

- Using the same guidelines as for the large sample

test for means (above) and the 5-step method,

work with a partner and try 9.11 to test for a

difference in proportions. - The answer to this question can be found at the

back of your text.

Formula (t-test) for Hypothesis Testing with

Sample Means (Small Samples N1 N2 lt 100)

FormulaS.D Note Use t-table with df

N1 N2 - 2

Example

- Using the same format as for the large sample

test (above) and the 5-step method, work with a

partner and try 9.7a - The answer to this question can be found at the

back of your text.

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