Z test for proportions

• Assumptions

Method of simple random sampling is used

There are only two outcomes, p (probability of success) or q (probability of failure). Also, p + q = 1

If np≥10 and nq≥10, then normal approximation to binomial is valid

• 5 step Hypothesis

Hypothesis

Ho: population proportion is equal to po. p = po

Ha1: population mean is equal to po. p≠po  (Two tailed test)

Ha2: population mean is equal to po. p<po  (Left tailed test)

Ha3: population mean is equal to po. p>po  (Right tailed test)

Here p_o takes a particular value.

 Test Statistic

 Critical value

-z(a/2), z(a/2) (Two tailed test)

-z(a) (Left tailed test)

z(a) (Right tailed test)

 P-value

2*(1-P(Z≤|z|)  (Two tailed test)

P(Z≤z)  (Left tailed test)

P(Z≥z)  (Right tailed test)

 Decision rule

Reject Ho if |z|>z(a/2) or p-value < alpha (two tailed test)

Reject Ho if –z < -z(a) or p-value < alpha (left tailed test)

Reject Ho if z > z(a) or p-value < alpha (right tailed test)

• Confidence Interval

Standard error (SE) and margin of error (ME) is given by:

100(1-alpha)% Confidence interval for population mean is given by:

This implies I am 100(1-alpha)% confident that estimated population proportion lies in the obtained interval.

One sample z test for Proportions CALCULATOR