# Independent Samples t-Test: Chips Ahoy® vs. Supermarket Brand

as part of its collaboration with the SERC Pedagogic Service.

## Summary

In this hands-on activity, students count the number of chips in cookies in order to carry out an independent samples t-test to see if Chips Ahoy® cookies have a higher, lower, or different mean number of chips per cookie than a supermarket brand. First there is a class discussion that can include concepts about random samples, independence of samples, recently covered tests, comparing two parameters with null and alternative hypotheses, what it means to be a chip in a cookie, how to break up the cookies to count chips, and of course a class consensus on the hypotheses to be tested. Second the students count the number of chips in a one cookie from each brand, and report their observations to the instructor. Third, the instructor develops the independent sample t-test statistic. Fourth, the students carry out (individually or as a class) the hypothesis test, checking the assumptions on sample-size/population-shape.

## Learning Goals

- Reinforce concepts of (simple) random samples.
- Introduce the concept of independent samples (vs. paired samples, if appropriate).
- Introduce null and alternative hypotheses for comparing two parameters, and the different ways to write them down (i.e., mu1 = mu2 vs. mu1 – mu2 = 0).
- Address the importance of having uniform guidelines for recording data, and possible, the learning effect.

## Context for Use

- Can be used at any educational level, or any course, that covers the independent samples t-test (or t- confidence interval).
- It can be inserted in any existing format for presenting the independent samples t-test.
- The actual counting of the chips and recording the data (classroom computer) may take 15 minutes, depending on the class size.
- The activity is effective with classes of 20-25 students, but
- in small classes students can count chips in multiple cookies, and
- the number of cookies you need for each sample can be dependent on your texts conditions for the independent samples t-test!

## Description and Teaching Materials

- Requires the following materials:
- One package of Chips Ahoy® cookies (approx 44 cookies) and one package of a store brand of chocolate chip cookies (I use Acme; the number of cookies in a bag will vary with the store brand).
- One, small paper plate per student, to break the cookies over. (Little paper plates usually come 100 to a pack. They require separating before class, because pairs of them stick together.)
- Wastebasket in the classroom.
- Napkins are optional.
- Requires statistical software or a statistically capable graphing calculator for recording the data, making appropriate graphs to check assumptions, and analyzing the data.
- Works the following way:
- Use your usual introduction or class discussion of comparing two parameters with null and alternative hypotheses. It can include concepts about random samples, independence of samples, ‘recently’ covered tests.
- A more specific discussion of "how should we" gather data in order to compare the mean number of chips per cookies for Chips Ahoy® and the supermarket brand, including, what it means to be a chip in a cookie, how to break up the cookies to count chips, what the students think the relationship is between the means, and a class consensus on the hypotheses to be tested. If students think there will be learning going on when they count the chips, then you should randomize the order in which they count chips.
- Hand out one paper plate to each student.
- Hand out one of each kind of cookie to each student. If they are not distinguishable, then give out the second cookie only after a student is finished counting the chips in the first. Give 'half' of the students Chips Ahoy® first if the class feels there is a learning effect (counting chips).
- When a student has the number of chips in each cookie (they can write the numbers on the paper plate), he or she brings the information to the front of the room where you can record it.
- Getting student help in c), d) & e) makes the process go faster.
- Return to your usual development of the independent samples t-test, but now with interesting data.
- More detailed instructions on doing the activity are found in the MSWord file Detailed Instructions for Chips Ahoy (Word) (Microsoft Word 31kB May17 07) and PDF file Detailed Instructions for Chips Ahoy (PDF) (Acrobat (PDF) 68kB May17 07). It includes the Adaptations and Warnings in the next section.

## Teaching Notes and Tips

- Context. This activity is meant to provide a context for discussing the independent samples t-test. Adapt it to your method of introducing the independent samples t-test, whether you cover the pooled, approximate, or both.
- Assumptions/conditions. The concept of comparing chips gives numerous points for discussion. If some assumptions fail, be prepared to "continue for the sake of illustration" and write a caveat.
- Independent random samples. Ask the students if you can consider the bags of cookies as randomly chosen? Are the two samples independent? What are the populations from which the samples came?
- Sample size conditions. If the populations of the number of chips are normal, we can use the procedures. Histograms and normal probability plots can be used to verify or eliminate normality. Of course if the sample sizes are large enough, the sample means are normal, but these conditions differ from text to text! For instance in some books the conditions are more robust, and depend on the sum of the sample sizes (Moore and McCabe), whereas in others the sample sizes must individually be at least 30 (say). Hence the size of the class needed to make the activity work well may depend on the text you use.
- Population variances equal? If you use the pooled t-test the samples should satisfy the conditions. Consider switching to the approximate t-test if you have not already!
- Warnings.
- If you teach in a computer lab, you can have the students move outside to break up the cookies. Crumbs in the keyboard may disqualify you from using the lab/classroom again.
- Be sure to point out that although the sample sizes may be the same for the two samples, this is not a paired t-test.

- Adaptations.
- The counting can be done in a take-home fashion. Give the students one of each cookie to take home in little plastic bags, adequately marking the cookies so that the data is not mixed up (wrap one in a piece of paper or little envelope, use markers, or see below).
- Because this is not a paired t-test, you can have some students do two of one kind of cookie, so that you get different sample sizes.
- Having extra cookies (and milk) for consumption is optional, because the counting is a destructive process.
- If you are independently wealthy or you school has a large budget, you can compare Chips Ahoy??? to a regional bakery's "gourmet" brand. (In at least New Jersey, we could choose Entenmann's!)

## Assessment

**Formative.**Consider the Chips Ahoy® activity from class, with its discussion of the hypotheses for the independent samples t-test, as well as its assumtions/conditions.

- Choose one of the following (that best fits your feelings):
- It helped me learn the independent samples t-test.
- It hindered me in my learning of the independent samples t-test.
- It neither helped me nor hindered me in my learning of the independent samples t-test.

- Explain what one or two aspects of the activity that led you to choose your answer.

**Summative.**(To be administered after discussion of comparing two proportions.) You and a classmate have been discussing the colors of Swedish fish and gummy bears. You think that the proportion of red in the Swedish fish is greater than the proportion of red in gummy bears, and your classmate thinks that they are the same (for a particular brand (instructors can insert a name)).

- Define the appropriate parameters, and write down the null and alternative hypotheses that you should be testing.
- Your classmate has suggested that you go to the supermarket and buy a package of each candy and use the data from the two packages to perform the hypothesis test. Comment on the validity of the assumptions for the appropriate test.
- After buying several packages (randomly chosen) of each kind of candy, you find that 23 of 75 Swedish fish are red, but 15 of 60 gummy bears are red. Comment of the validity of the sample size assumptions for the appropriate test.
- Complete the appropriate test, being sure to give all of the appropriate conclusions. [Instructor: we are looking for the so called "statistical conclusion" and the "real-world" conclusions.]

## References and Resources

Warner, Brad, and Rutledge, Jim, "Checking the Chips Ahoy! Guarantee," Chance Vol. 12, No. 1, Winter 1999, pp. 10-14.