6.6 Concluding a Test for a Population Proportion

Now the hard work is done. You have written your hypotheses, checked your conditions, calculated your statistics, but now what? What does this mean? 🤷🤷

Our basic conclusion hinges on one of two outcomes: you are either going to reject your null or fail to reject your null. These are based off of the probability of obtaining our test statistic.

❇️Important Note: NEVER ACCEPT YOUR Ho or Ha! ALWAYS REJECT/FAIL TO REJECT!**

The first, and most common way to conclude our significance test is using our p-value that is generated by our calculator. Remember, our p-value is the probability of obtaining our sample if we have a normal sampling distribution with the null value as our center. We conclude by comparing our p-value to our significance level (which is usually 0.05 unless otherwise noted). If our p-value is lower than our significance level (or our 𝞪), this means that it is unlikely to occur by random chance. Therefore, we have reason to reject our Ho.

If our p-value is not lower than our significance level (or alpha level), then we fail to reject our Ho (not accept). This means that we do not have evidence to reject our Ho in favor of our Ha, but we also don't have evidence to completely accept our Ho as fact.

While a p-value is the most common way to conclude a test, we can also use our z-score to conclude a test. Remember that a z-score is how many standard deviations we are above/below the mean. Therefore, if we have a z-score higher than 2, it is pretty unlikely to occur by natural chance (since we have checked our normal condition and know that we are dealing with a normal sampling distribution). This comes from the Empirical Rule that states that 95% of our data in a normal distribution falls within 2 standard deviations. Therefore, a z-score higher than 2 (or lower than -2) signifies that the probability of it occurring is likely less than 0.05. So we can conclude the same way as we did above with a p-value. It is especially easy to make a reject Ho decision when our z-score is really large (like 4+ or -4-). If our z-score is in the range of -2 to 2, it is really hard to reject our Ho, so we will likely fail to reject without further information.

Here is the template you can follow when concluding a one proportion z test for population proportion (or a 1-Prop Z Test):

"Since (p value)</>(alpha level), we reject/fail to reject
our Ho. We have/do not have significant evidence of ________(our Ha in context)."

The big three things you need to have in your conclusion to maximize our credit are:

🎥Watch: AP Stats - Inference: Hypothesis Tests for Proportions