OpenIntroOrg 2-Proportion Hypothesis Test and Confidence Intervals using Casio fx-9750GII

Learn how to perform two-proportion hypothesis tests and construct two-proportion confidence intervals on a Casio 9750 graphing calculator.

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Approximate audio script:

2-proportion Z Hypothesis Test and Confidence Interval

We'll cover how to conduct a hypothesis test for comparing two proportions, and we'll cover how to construct a confidence interval for the difference of two proportions. If you haven't already, please watch the 1-Proportion video first.

Context

We'll consider whether views changed on individual healthcare costs due to Obamacare from June 2009 to December 2013 [ref]. We might wonder, has the fraction of people who believe it will increase their healthcare costs changed? We can set up hypothesis for this question, and we'll use a significance level of alpha equals 0.01.

Back in 2009, a CNN poll of about 1,000 people found that 54% of US adults believed that Obamacare would increase their healthcare costs, or 540 out of 1000 respondents. In 2013, a CNN poll of 1,035 US adults found the number was 64%, or 662 out of 1,035 respondents. We've verified the conditions for this sample are satisfied so that we can move ahead with conducting the hypothesis test.

Hypothesis test

To conduct a hypothesis test using our Casio calculator,

go to Menu,
then to Stat,
choose F3 for hypothesis test,
then Z, and finally,
select 2-P for 2-proportions.

We'll be testing whether there is any difference, so we want a not-equals sign, which is already shown here. Next, we'll need to enter the data for each sample. To get x1, we had 54% of 1000 people, so 540, and we had 1000 for n1. For x2 we have 64% of 1035, so 662, and then n2 is 1035. Finally, hitting execute will run the numbers. We can see that

the alternative hypothesis was 2-sided, as we had specified,
the Z test statistic is -4.57,

the p-value is 4.9e-06, which is written in scientific notation. We can write the long form by writing 4.9 and then moving the decimal 6 places to the left since it is E negative 6.

The sample proportions are 0.54 and 0.64, and
the pooled proportion is 0.59.

Since the p-value is smaller than alpha = 0.01, we reject the null hypothesis. That is, we have strong evidence that the proportion of US adults who believe Obamacare will increase their healthcare costs was higher in 2013 than in 2009. Notice that, even though the alternative hypothesis was 2-sided, if we reject the null hypothesis, we can conclude the direction based on the data.

When I'm done, I can exit out to the main Stat page.

Confidence interval

We can also construct a confidence interval for the increase in the fraction of adults who believe Obamacare will increase their healthcare costs.

Go to INTR, for "confidence interval",
choose F1 for Z,
then choose 2-P for 2-proportions.

We'll use a 99% confidence level, so we'll enter 0.99 for the confidence level. All of our data has carried over from the hypothesis test, so we can now hit execute to get the interval.

We have a left end of the interval of -0.156, a right end of -0.044, and a point estimate of each proportion of 0.54 and 0.64. Notice that the interval is actually reversed from what we want. We observed an increase of 10% from 2009 to 2013, so the actual interval should be 0.044 to 0.156. It is common to have the interval get turned around in the calculator, which is why it is important to read your calculator results carefully and think about how it fits in the context of the problem. That is, always sanity check your work.

Finally, we should always put a confidence interval in context: we are 99% confident that there was a 4.4% to 15.6% increase between 2009 and 2013 in the fraction of US adults who believed Obamacare would increase their healthcare costs.

2-Proportion Hypothesis Test and Confidence Intervals using Casio fx-9750GII
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