Hypothesis Testing: One Proportion


As the name implies, one proportion tests deal with situations in which you are analyzing a single proportion. A few symbols need to be defined before we dive in:

  • \(p_o\) refers to the null proportion, or the given proportion that is assumed to be true.
  • \(q_o\) refers to 1 minus the null proportion, i.e. \(1 - p_o\)
  • p-hat ( \(\widehat {p}\)) refers to the sample proportion that you will use to disprove the null. 
  • \(n\) refers to the sample size



With a fair coin, the proportion of heads is 0.5. However, when you flip your coin 100 times, you only get 20 heads (making a proportion of 0.2). Is this data sufficient to prove that your coin is not fair?


Step 1: Name Test: 1-Proportion Hypothesis Test

Step 2Define Test:

H0:  p = 0.5

HA:  p < 0.5

Step 3: Assume \(H_0\) is true and define its normal distribution. Then check the conditions.

1. The data is from a simple random sample.

2.  \(N > 10n\)

3.  \(np_o > 10 \) and  \(nq_o > 10 \)

Step 4: Using the normal distribution, calculate the test statistics and p-value.

Test Statistic (1 Proportion):  \(z = {\widehat{p} - p_o \over \sqrt{p_o q_o \over n}}\)

Test Statistic: 

 \(z = {0.2 -0.5 \over \sqrt{(0.5)(0.5) \over 100}}\)  →   \(z = -6\) 


The p-value will be found by using the normal cdf function on your calculator: 

  • lower limit: -999
  • upper limit: \(z\) 
  • distribution center: 0
  • standard deviation: 1
  • All together, it looks like this: normalcdf (-999, \(z\), 0, 1)

*Note: If it was a right-sided test and the test statistic was positive (z > 0), then your lower limit would be the test statistic (z) and your upper limit would be 999.

In this case, we do normalcdf (-999, -6, 0, 1) to get a p-value of 9.9 \(\times\) 10-10 (approximately zero).

Step 5: Analyze your results and determine if they are statistically significant.

We calculated a p-value of approximately zero. This p-value is less than the assumed significance level of 0.05. Therefore, we reject the null hypothesis. The data supports the claim that the proportion of heads is less than 0.5.