** Z
Alpha Over Two** (Z

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

1) Use the normal distribution table (Table A-2 pp.724-25).

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

90% written as a decimal is 0.90.

1 – 0.90 = 0.10 = α and α/2 = 0.10/2 = 0.05.

Look for 0.05 = 0.0500 or two numbers surrounding it in the body of Table A-2

(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 724.

The number 0.0500 is not in the table, but it is between 0.0505 and 0.0495, which are in the table.

Next, check the differences between these last two numbers and 0.0500 to see which number

is closer to 0.0500. 0.0505 – 0.0500 = 0.0005 and 0.0500 – 0.0495 = 0.0005.

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

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

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

(-1.64 + (-1.65))/2 = -1.645

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

2)
Use the *t*-Distribution
table (Table A-3, p. 726).

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

98% written as a decimal is 0.98.

1 – 0.98 = 0.02 = a and α/2 = 0.01.

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

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

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

3) Use the TI 83/84 Calculator.

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

99% written as a decimal is 0.99.

1 – 0.99 = 0.01 = α and α/2 = 0.005.

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_{α/2}.

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

4)
Memorize the values of Z_{α/2}.

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

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

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

Confidence (1–α) g 100% |
Significance α |
Critical Value Z |

90% |
0.10 |
1.645 |

95% |
0.05 |
1.960 |

98% |
0.02 |
2.326 |

99% |
0.01 |
2.576 |