In short, if the absolute value of the T is higher than 1.96, then one may

conclude (with 95% Confidence) that the means of the samples differ by

the hypothesized difference.

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Chapter 11: Hypothesis Testing

(b) One“tailed (left-tail)

The hypothesis:

” H0 (Null Hypothesis): u1” u2 >= 0

” Ha (Alternate hypothesis): u1” u2 < 0 (one“tailed)

Critical region:

” “Fail to accept” the null hypothesis if the value of the calculated

T is lower than ““1.64.” Examples of such T values are: ““2.12”

and ““1.78.”

” “Fail to reject” the null hypothesis if the absolute value of the

calculated T is greater than ““1.64.” Examples of such T values

are: “+1.78” and “0.00.”

In short, if the T is lower than ““1.64,” one may conclude (with 95%

Confidence) that the means of the samples differ by the hypothesized

difference.

(c) One“tailed (right-tail)

The hypothesis:

” H0 (Null Hypothesis): u1” u2 <= 0

” Ha (Alternate hypothesis): u1” u2 > 0 (one“tailed)

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Statistical Analysis with Excel

Critical region:

” “Fail to accept” the null hypothesis if the value of the calculated

T is greater than “+1.64.” Examples of such T values are: “+2.12”

and “+1.78.”

” “Fail to reject” the null hypothesis if the absolute value of the

calculated T is less than “+1.64.” Examples of such T values are:

““1.78” and “0.00.”

In short, if the T is greater than “+1.64,” then one may conclude (with 95%

Confidence) that the means of the samples differ by the hypothesized

difference.

Go to the menu option TOOLS/DATA ANALYSIS22. Select the option “T-

test: Two“Sample Assuming Unequal Variances.” The next table shows a

sample output23 for a T-test assuming unequal variances.

Table 34: Output of Two Sample T-test (assuming unequal variances)

s1 s2

If you do not see this option, then use TOOLS / ADD-INS to activate the Add-In for

22

data analysis. Refer to section 41.4.

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

23

My experience is that many readers glaze over the examples and do not go through

the difficult step of drawing inferences from a result if the sample data results are

the same as those in the examples in the book.

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Chapter 11: Hypothesis Testing

s1 s2

Mean 7.32 7.23

Variance 32.68 40.13

Observations 168 168

Hypothesized Mean Difference 5

Df 331

T Stat “7.465

P (T< = t) one“tail 3.72E“13

T Critical one“tail 1.649

P (T< = t) two“tail 7.43E“13

T Critical two tail .967

Interpreting the output

The row “Mean” shows the estimated means for the two samples s1 and

s2. The next column “Variance” displays the calculated variance for these

sample mean values. “Df” shows the “Degree of Freedom.” The degrees of

freedom equal the total sample points (the sum of the sample sizes of the

two samples) minus the one degree of freedom to account for the one

equation (the “hypothesized mean difference” which here is “u1 ” u2 = 5”)

. So, degrees of freedom equals “168 + 168 -1 = 331”.

(a) Two“tailed