Module 8: Inference for One Proportion
Putting It Together: Inference for One Proportion
Putting It Together: Inference for One Proportion
Let’s Summarize
In Inference for One Proportion, we learned two inference procedures to draw conclusions about a population proportion:
- A confidence interval when our goal is to estimate a population proportion.
- A hypothesis test when our goal is to test a claim about a population proportion.
Confidence Interval for Estimating a Population Proportion
- A confidence interval estimates the population proportion with a range of possible values. The interval is based on a sample proportion and a margin of error.
- Every confidence interval has a confidence level associated with it. The confidence level is a probability statement. It tells us the chance that a confidence interval, with a specific margin of error, contains the population proportion. But we can never determine if a specific interval does or does not contain the population proportion. We also cannot determine the probability that the population proportion lies in a specific interval. We can only say that in the long run the confidence level describes the percentage of the confidence intervals that will estimate the population proportion within a specific margin of error.
- We can calculate a confidence interval for a population proportion when we can use a normal distribution to model the long-run behavior of sample proportions. We can use a normal distribution model when there are at least 10 observed successes and 10 observed failures.
- We calculate the confidence interval for a population proportion using this formula:
[latex]\text{Sample proportion} \pm \text{margin of error}[/latex]
[latex]\hat{p} \pm Zc\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}[/latex]
where Zc depends on the confidence level. The part of the formula after the ± is the margin of error. The most common confidence levels are 90%, 95%, and 99%. The critical z-scores are 1.65, 1.96, and 2.576.
- The margin of error comes from the standard error in the sampling distribution. Sample proportions from larger sample sizes have less variability, so the standard error is smaller. Therefore, confidence intervals based on larger sample sizes will have a smaller margin of error. This fits our intuition that larger samples will give more accurate estimates of the population proportion.
- A higher level of confidence makes us more confident that the interval contains the population proportion because the interval is wider. This also means that the margin of error is larger.
Hypothesis Tests in General:
Hypothesis tests consist of four steps, which apply to all the hypothesis tests we will do in this course.
Step 1: Determine the hypotheses.
The hypotheses are statements about the parameter(s) in question. The null hypothesis, H0, is always a statement of equality and usually means no change or difference. The alternative hypothesis, Ha, is always an inequality, either <, >, or ≠, and is based o