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Z Alpha Over Two (Za/2)

There are four ways to obtain the values needed for  Za/2 :

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

Example  Find Za/2  for 90% confidence.

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

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.

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.

Thus Za/2 = 1.645  for 90% confidence.

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

Example  Find  Za/2  for 98% confidence.

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

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

Za/2  is the very last entry in the column under 0.01.

Hence  Za/2 = 2.326  for 98% confidence.

iii)         Use the TI 83/84 Calculator.

Example  Find  Za/2  for 99% confidence.

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

            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  Za/2 .

            Consequently,  Za/2 = 2.576  for 99% confidence.

iv)        Memorize the values of  Za/2 .

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

and the values of  Za/2  corresponding to these confidence levels are always the same.

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

Confidence

(1a)×100%

Significance

a

Critical Value

Za/2

90%

0.10

1.645

95%

0.05

1.960

98%

0.02

2.326

99%

0.01

2.576

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