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

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

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Approximate script:

We'll cover how to run a hypothesis test and construct a confidence interval for a difference of two means using a Casio fx-9750 graphing calculator in the context of an example.

If you haven't already watched the 1-sample inference video for this Casio calculator, please do so!

Do embryonic stem cells (ESCs) help improve heart function following a heart attack? We'll consider an experiment where 18 sheep that had heart attacks were randomized into a treatment and control group. Sheep in the treatment group received embryonic stem cell treatment, while sheep in the control group received no special treatment. the heart pumping capacity of the sheep's hearts were measured before and after the experiment, and we'll analyze the relative changes in heart pumping function, shown here. While we can view the raw data for such a small data set, it's still helpful to graph the data. Importantly, we don't see any extreme outliers or evident skew in the data.

Hypothesis test

We want to evaluate whether embryonic stem cells had an influence on the change in heart pumping function of sheep that had suffered a heart attack, we first write down hypotheses. We'll use a one-sided test and a significance level of 0.05.

To conduct the hypothesis test, navigate to the the Stat section. Since we don't have summary data, we'll need to enter the raw data into lists. Here I'll enter the control data into L1 and the treatment data into L2. With the data entered, I'll enter into TEST for hypothesis test, then T for t test, and I'll choose 2-S for 2 samples of numerical data.

I start by indicating that I'll use lists for the data in the test. Next, I specify the sidedness of the hypotheses. I'll think of mu 1 as representing the control group and mu 2 as representing the treatment group, so I'll choose less than for the sidedness of the test. Next, I need to specify the lists. These happen to be the correct lists already. The frequency values are for if we have tabulated data. Usually that's not the case, so a value of 1 is preferred. Finally, I choose whether I want to pool the data or not for the standard deviations. It's common practice to not make the assumption that the standard deviations are equal and so not pool for the standard deviation calculation.

It is important that I've specified the proper lists for the proper groups, especially when doing a one-sided test. Recall that I entered the control data in List 1 and I decided that mu 1 would represent the true average for the control group. The treatment group used List 2 and mu 2. Always keep your data organized to avoid silly mistakes, and always sanity check your work.

I can hit execute to get the test results. First up is a summary of the alternative hypothesis, then the test statistic, the p-value of 8.4e-04, which we can write as 0.00084, and the degrees of freedom for the test statistic. Because the p-value is so small, we reject the null hypothesis and conclude that the embryonic stem cells causes an improvement in the heart pumping function for sheep that have had a heart attack relative to sheep that receive no special treatment.

Now that I've completed the hypothesis test, I can exit out to the main Stat page.

Confidence intervals

Since there was a statistically significant difference, it makes me wonder, how big was that difference? I'd like to answer that question with a confidence interval. I'll again use the data from the lists, and I'll navigate to INTR for confidence interval. I want to construct a T interval, and I want to do so in a 2-sample setting.

First, I specify whether to use lists of data or summary information, then I choose the confidence level, and finally I specify the data. Here again, we have the option of whether to pool or not. The values provided all match to what I want. Hitting execute, I can see the confidence interval.

The left end of the interval is -11.3 and the right end is -4.4. However, something seems weird: this interval is entirely below zero, but we observed an increase in heart pumping function in the treatment group. Instead, our interval is the reverse effect. To get the proper interval, we can either go back and reverse the lists, or we can take negatives of the interval endpoints and reverse the endpoints to get the confidence interval of interest.

We are 90% confident that embryonic stem cells increase the heart pumping function of sheep that have had a heart attack by 4.4 to 11.3 percent relative to no special treatment. Notice that this interval is entirely above 0, so it is consistent with the conclusion of our hypothesis test.
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2-Sample Hypothesis Test and Confidence Intervals using Casio fx-9750GII
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