also

z-test for a Single Proportion

Margin of Error for a Proportion

margin of error = s_p*CV_z

For example, the margin of error for a 95% confidence interval for p = .5 with 1000 observations is,

Confidence Interval for a Single Proportion

Single Sample z-test Example

A coin is tossed 4040 times with 2048 heads. Is it a fair coin?

H₀: p = 0.5       H₁: p <> 0.5

Use a two-tail test with a = 0.05.

Critical Value of z = ±1.96

phat = 2048/4040 = 0.5069       qhat = 1 - 0.5069 = 0.4931

Standard Error = SQRT((.5069)(.4931)/4040) = 0.0079

Observed z = (0.5069 - 0.5000)/0.0079 = 0.88

Decision: Fail to reject H₀.

The data do not support the hypothesis that the coin is biased.

Stata Example

prtesti 4040 .5069 .5

One-sample test of proportion                      x: Number of obs =     4040
------------------------------------------------------------------------------
    Variable |       Mean   Std. Err.                     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |      .5069   .0078657                      .4914835    .5223165
------------------------------------------------------------------------------
    p = proportion(x)                                             z =   0.8771
Ho: p = 0.5

     Ha: p < 0.5                 Ha: p != 0.5                   Ha: p > 0.5
 Pr(Z < z) = 0.8098         Pr(|Z| > |z|) = 0.3804          Pr(Z > z) = 0.1902

Construct a 95% confidence interval on the preceding example.

phat ± CV_z*standard error
0.5069 ± 1.96*0.0079 = 0.5069 ± 0.0155
(0.4914, 0.5224)

Since the confidence interval contains the hypothesized population proportion, fail to reject the H₀.

Stata Example

cii 4040 2048

                                                   -- Binomial Exact --
Variable |     Obs         Mean    Std. Err.       [95% Conf. Interval]
---------+-------------------------------------------------------------
         |    4040     .5069307    .0078657        .4913908    .5224614

z-test for Comparing Two Independent Proportions

Two Independent Sample z-test Example

In a survey of high school seniors, how many have taken any AP math classes.

Group	n	Frequency	p
Urban	261	127	0.487
Rural	160	65	0.400

Is the proportion of urban seniors significantly different from rural seniors?

H0: p₁ = p₂       H₁: p₁ <> p₂

Use a two=tailed test at a = 0.05

Critical Value of z = ±1.96

phat = (127+65)/(261+160) = 192/421 = 0.456
qhat = 1 - phat = 1 - 0.456 = .544

s_p = SQRT(0.456*0.544*(1/261 + 1/160)) = 0.05

z = (0.487 - 0.400)/0.05 = 0.087/0.5 = 1.74

Decision: Fail to reject H₀

There is no significant difference between the two proportions.

Stata Example

prtesti 261 .487 160 .4

Two-sample test of proportion                      x: Number of obs =      261
                                                   y: Number of obs =      160
------------------------------------------------------------------------------
    Variable |       Mean   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |       .487   .0309388                      .4263611    .5476389
           y |         .4   .0387298                      .3240909    .4759091
-------------+----------------------------------------------------------------
        diff |       .087   .0495702                     -.0101558    .1841558
             |  under Ho:   .0499896     1.74   0.082
------------------------------------------------------------------------------
        diff = prop(x) - prop(y)                                  z =   1.7404
    Ho: diff = 0

    Ha: diff < 0                 Ha: diff != 0                 Ha: diff > 0
 Pr(Z < z) = 0.9591         Pr(|Z| < |z|) = 0.0818          Pr(Z > z) = 0.0409

Confidence Interval Example

In a survey of high school seniors, how many have taken any AP math classes.

Group	n	Frequency	p
Urban	261	127	0.487
Rural	160	65	0.400

Construct a 95% confidence interval for the difference in these two independent proportions.

The critical value of z at a = 0.05 is ±1.96

s_D = SQRT((0.487 * 0.513)/261 + (0.400 * 0.600)/160)) = 0.0496

D = 0.487 - 0.400 = 0.087

D ± CV_z * s_D = 0.087 ± 1.96 * 0.0496 = 0.087 ± 0.097 =
(-0.01,    0.184)

For hypothesis testing purposes, see if 0 is in the interval. It is in the interval, thus fail to reject the H₀.

There are no differences in the two proportions beyond chance.

prtesti 261 .487 160 .4

Two-sample test of proportion                      x: Number of obs =      261
                                                   y: Number of obs =      160
------------------------------------------------------------------------------
    Variable |       Mean   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |       .487   .0309388                      .4263611    .5476389
           y |         .4   .0387298                      .3240909    .4759091
-------------+----------------------------------------------------------------
        diff |       .087   .0495702                     -.0101558    .1841558
             |  under Ho:   .0499896     1.74   0.082
------------------------------------------------------------------------------
        diff = prop(x) - prop(y)                                  z =   1.7404
    Ho: diff = 0

    Ha: diff < 0                 Ha: diff != 0                 Ha: diff > 0
 Pr(Z < z) = 0.9591         Pr(|Z| < |z|) = 0.0818          Pr(Z > z) = 0.0409

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