** Z Alpha Over Two** (Z

There are four ways to obtain the
values needed for Z_{a/2}
:

i) Use the normal distribution table (Table IV pp. A10-A11).

*Example* Find Z_{a/2 }for 90% confidence.

90% as a decimal is 0.90 and 1 – 0.90 = 0.10 = a .

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Look for 0.05 = 0.0500 or two numbers surrounding it in the body of Table IV

(i.e. below the first row and to the right of the first column).

Since 0.0500 is less than 0.5, we look on page A10.

0.0500 itself is not in the table, but it is between 0.0505 and 0.0495.

Check the differences to see which one is closer to 0.0500.

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Since the differences are equal, we average the corresponding standard scores.

0.0505 is to the right of –1.6 and under 0.04, its standard score is –1.64.

0.0495 is to the right of –1.6 and under 0.05, its standard score is –1.65.

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Thus Z_{a/2
}= 1.645 for 90% confidence.

ii)
Use the *t*-Distribution table (Table VI, p. A12).

*Example* Find Z_{a/2} for 98% confidence.

98% as a decimal is 0.98 and 1 – 0.98 = 0.02 = a .

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Find 0.01 in the “**df**” row at the top of page A12.

Z_{a/2} is the very last entry in the column under
0.01.

Hence Z_{a/2}
= 2.326 for 98% confidence.

iii) Use the TI 83/84 Calculator.

*Example* Find Z_{a/2} for 99% confidence.

99% as a decimal is 0.99 and 1 – 0.99 = 0.01 = a .

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Press “2ND” and “VARS” on your TI 83/84 calculator.

Choose “invNorm(” and press “ENTER”.

You should see “invNorm(” on your calculator screen.

Type in 0.005, add a right parenthesis and press the “ENTER” key.

The result,
rounded to three decimal places, is the opposite of Z_{a/2}
.

Consequently, Z_{a/2}
= 2.576 for 99% confidence.

iv) Memorize
the values of Z_{a/2}
.

The only confidence levels we use on tests or assignments are 90%, 95%, 98% and 99%,

and the values of Z_{a/2} corresponding to these confidence levels are
always the same.

As a result, memorizing the necessary values of Z_{a/2}
is fairly easy to do.

Confidence (1–a)×100% |
Significance a |
Critical Value Z |

90% |
0.10 |
1.645 |

95% |
0.05 |
1.960 |

98% |
0.02 |
2.326 |

99% |
0.01 |
2.576 |