” H0 (Null Hypothesis): u1” u2 = 5

” Ha (Alternate hypothesis): u1” u2 <> 5, where u1 is the mean

of sample s1 and u2 the mean of sample s2.

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Statistical Analysis with Excel

The calculated T statistic is ““7.465.” The P value for the two“tailed test

is “3.72 multiplied by the 13th point after the decimal” or

“0.000000000000372.” As the P value is less than 0.01, the hypothesis is

“significant24“at the 99% Confidence level or “alpha = 0.01” level of

significance. (The natural extension of this inference is that the

hypothesis is significant at the 95% and 90% Confidence levels also.)

The region for the two“tailed test is “> 1.967 or < “1.967.” In this

example, the test is significant (at a 0.05 level of significance because the

estimated T lies in the critical region. (The estimated T of ““7.465” lies in

the region “< “1.967”.)

(b) One“tailed (left-tail)

The hypothesis was:

” H0 (Null Hypothesis): u1” u2 >= 5

” Ha (Alternate hypothesis): u1” u2 < 5, where u1 is the mean of

sample s1 and u2 the mean of sample s2.

The P value for the one“tailed test is “7.45 multiplied by the 13th point

after the decimal” or “0.000000000000745.” The relevant test here is the

left“tail because the T statistic is a negative value. As the P value is less

If a test is “significant” the implication is a “failure to accept” the null hypothesis.

24

The test T statistic lies in the critical region. In informal terms, the alternate

hypothesis is “correct.”

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Chapter 11: Hypothesis Testing

than 0.01, the hypothesis is “significant” at the 99% Confidence level or

“alpha = 0.01” level of significance. (The natural extension of this

inference is that the hypothesis is significant at the 95% and 90%

Confidence levels also.)

Another way to test the hypothesis is to compare the estimated T statistic

to the critical region shown in the column “T Critical one“tail.” The

region for the left“tailed test is “< “1.649”. In this example, the test is

“significant25“at a .05 level of significance because the estimated T lies in

the critical region. (The estimated T of ““7.465” lies in the region “< “

1.649”.)

(c) One“tailed (right-tail)

The hypothesis was:

” H0 (Null Hypothesis): u1” u2 <= 5

” Ha (Alternate hypothesis): u1” u2 > 5, where u1 is the mean of

sample s1 and u2 the mean of sample s2.

The region for the right“tailed test is “> 1.649”. In this example, the test

is not significant because the estimated T does not lie in the critical

region. (The estimated T of ““7.465” is not in the region “>1.649”.)

If a test is “significant” the implication is a “failure to accept” the null hypothesis.

25

The test T statistic lies in the critical region. In informal terms, the alternate

hypothesis is “correct.”

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Statistical Analysis with Excel

11.2.C T-TEST: TWO“SAMPLE ASSUMING EQUAL VARIANCES

This tool performs a two“sample student's T-test” under the assumption

that the variances of both data sets are equal. The hypothesis and

interpretation of results is the same as for the Two“Sample Assuming

Unequal Variances. (See previous sub-section).

The next table shows the result this type of test26.

PAIRED SAMPLE T-TESTS

11.3

This tool performs a paired two“sample T-test to deduce whether the

difference between the sample means is statistically distinct from a

hypothesized difference. This T-test form does not assume that the

variances of both populations are equal. You can use a paired test when

there is a natural pairing of observations in the samples, such as when a

sample group is tested twice” before and after an experiment. The tested

groups form a “Paired Sample” with the same respondents sampled

“before” and “after” an event.

Go to the menu option TOOLS/DATA ANALYSIS27. Select the option “T-

I do not supply the sample data for most of the examples in chapter 42 to chapter 46.

26