Learn Statistics Hypothesis Testing a Proportion

Testing a hypothesis about a population proportion involves determining whether a sample proportion differs significantly from a hypothesized population proportion. Here's a detailed guide to hypothesis testing for a proportion:

Steps for Hypothesis Testing a Proportion

1. Formulate Hypotheses:



• Null Hypothesis (H₀): 𝑝 = 𝑝 0 p=p 0


• The population proportion is equal to the hypothesized proportion.




• Alternative Hypothesis (H₁): 𝑝 ≠ 𝑝 0 p  =p 0 ​ or 𝑝 > 𝑝 0 p>p 0 ​ or 𝑝 < 𝑝 0 p< p 0


• The population proportion is not equal to, greater than, or less than the hypothesized proportion (depending on the test type).





2. Choose a Significance Level (α):



• Common values are 0.05, 0.01, or 0.10.



3. Select the Appropriate Test:



• Use the z-test for proportions.



4. Calculate the Test Statistic:



• Use the sample data to calculate the z-score.






5. Determine the p-value:

• The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, given that the null hypothesis is true.



6. Compare p-value with α:



• If p-value ≤ α, reject H₀.
• If p-value > α, do not reject H₀.



7. Draw a Conclusion:



Based on the comparison, conclude whether or not there is enough evidence to support the alternative hypothesis.

Example

Suppose we want to test if the proportion of people who support a new policy is different from 50%. We collect a random sample of 1000 people and find that 540 support the policy.

1. Formulate Hypotheses:



• H₀: 𝑝 = 0.50 p=0.50
• H₁: 𝑝 ≠ 0.50 p  =0.50 (two-tailed test)



2. Significance Level:



• α = 0.05



3. Sample Proportion:






4. Conclusion:



• There is significant evidence to conclude that the proportion of people who support the new policy is different from 50%.

Visual Representation

Here's an image illustrating this hypothesis test:

In the image:

• The blue curve represents the sampling distribution of the sample proportion under the null hypothesis.


• The red areas represent the rejection regions for a two-tailed test at α = 0.05.


• The vertical line represents the calculated z-value, showing it falls in the rejection region.