ISCAM

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Section 1 (9-10am)Section 2(10-11am)
Goals: In this lab, you will
- Learn how to distinguish independent sample study designs and paired sample study designs
- Consider advantages of paired sample study designs
- Learn how to correctly analyze paired data (simulation and theory-based)
Background

Student researchers (Coutin & Heffler, 2021) wanted to know whether listening to up-tempo music causes college students to tend to type faster. To collect their data (number of words typed correctly in one minute), the students planned to use the 60-second Easy-Text typing test (TypingTest.com). They recruited 34 college students from groups they were associated with on campus (e.g., athletic teams, musical groups). For the up-tempo music they selected Overture to Candide performed by the London Symphony Orchestra.

Let μnomusic represent the population mean typing speed without the music and μmusic the population mean typing seed with the music.

(a) State the student researchers’ null and alternative hypotheses in symbols and in words.
Symbols you might be able to copy and paste?

μ σ s n ≤ ≥ ≠ x̄ p̂ 𝑝̂

Study Design
(b) Describe a completely randomized design for conducting this study. Identify the experimental units and variables of interest. Classify the variables as explanatory and response, as well as quantitative or categorical.
(c) Identify some precautions you would take in carrying out this design.
Consider the following results for the number of words per minute by each participant.
(d) Calculate the difference in means
(e) Would a two-sample t-test likely be valid for these data? How are you deciding?
The one-sided p-value from a two-sample t-test is 0.2299, failing to provide convincing evidence that the average typing speed is faster with (this) music than the average typing speed without music. But maybe there is a genuine difference and our sample sizes are just too small to detect it.

Increasing the sample sizes would increase our power but if the actual difference in typing speeds is 4 words per minute on average, we would need sample sizes on the order of 100 participants, in each group, to have at least 80% power.
(f) Suggest another modification you could make to the experimental design (other than increasing the sample sizes) that would give you a better chance of detecting a difference in the average typing speeds between the two treatments if one exists? Be sure to explain why you believe this new design will be advantageous in detecting a difference between the two treatments. Also,
(g) How will randomness be used in this new study design and why is that important?

Paired Designs

Definition: In a paired design, rather than splitting the observational units into two distinct groups, they are paired in a way where we expect the observations within a pair to be more similar to each other than to observations in other pairs. This can explain some variation in the response variable.

For example, we can have each person take the typing test both with and without music. This allows us to compare the two typing speeds for each individual to each other, which should be very similar apart from the music, and to account for variation in typing speeds across individuals.

Quick Check: If you answer correctly, the next question will appear. (There are four total)
(h1) A farmer wants to see whether referring to cows by name increases their milk production. He selects half of his cows at random, gives them names, and frequently calls them by name. The other half of his cows he does not call by name. Then he measured the milk production of each cow over one week.

independent samplespaired samples
(h2) To compare the effectiveness of two different language study programs (e.g., Duolingo and Babbel), participants were assigned to enroll in a six-week summer session with each program, either Duolingo or Babbel first.

independent samplespaired samples
(h3) The weights of 10-year-olds in 2009 are being compared to the weights of 10-year-olds in 1994.

independent samplespaired samples
(h4) A farmer wants to know whether hand-milking or machine-milking tends to produce more milk from cows. He examines records of how much milk the cows have produced in the past, and order them from most to least productive. For the top two milk producers, randomly assign one to hand-milking and the other to machine-milking. Do the same for the next two and the next two and so on.

independent samplespaired samples

Analyzing Paired Data
Here are the results from the students' paired design.
(i) How many participants were involved in this study?
(j) Explain why it would NOT be appropriate to carry out either a randomization test (shuffling) or a two-sample t-test to compare the results with and without music for this study.
Instead, we can focus on the differences as the response variable.
(k) Summarize the behavior of these differences in context. (including clarification of whether this is (music - no music) or (no music - usci)
(l) How does the standard deviation of the differences compare to the standard deviations of the original groups? Explain why this difference makes sense in context.

Analyzing Paired Data - Simulation

In the Matched Pairs Randomization applet (open in larger window?)

Clear the Paste Paired data window
Type: TypingMusic.txt.
Press Use Data (twice).

The dotplots (Scroll right) should then show both sets of data, connecting the paired observations, and their differences.

Check the Randomize box and press Randomize.

For each pair, the applet will virtually “flip a coin” and if the coin lands heads, the two observations for that person will change positions. The new dotplots and the new set of differences for these rearranged values will be displayed. The mean of these differences will appear in the bottom dotplot.

Uncheck Animate
Press Randomize four more times to get a sense of the variability in the mean difference from repetition to repetition.
Change the number of repetitions from 1 to 995 (for a total of 1000) and press Randomize. (Scroll to the right.)

(m) Explain what distribution is being displayed in the bottom (grey) dotplot (what order of subtraction did the applet use?).
(n) Where is the mean of the distribution of the average differences? Why should you expect that?
(o) How surprising does our observed value for the mean difference appear to be, under the simulation’s assumption that presence/absence of up-tempo music does not affect typing speed?
(p) Use the applet to determine the proportion of simulated Average Differences that are more extreme than what we observed, and report the empirical p-value. [Hint: Be sure to consider the alternative hypothesis when deciding what to consider as “more extreme.”]
Include a screen capture of the output.

Cancelof

Analyzing Paired Data - Mathematical Model (Theory-Based)
Return to your results on the previous page or regenerate a null distribution. Or can use a second computer, you don't need to make a screen capture.
(q) If you change the Statistic from Avg Difference to t-statistic (and check the Overlay t distribution box), do the standardized statistics appear well-modeled by a t-distribution?
(r) Use the summary statistics (of the differences) and the equation above to calculate, by hand (show your work), and then interpret the value of this standardized statistic.
(s) Checking the Overlay t distribution box in the applet, how does the p-value compare to the empirical p-value from the simulated paired-randomization test? Based on this comparison, do the t-procedures appear to be valid for these data (the differences)?
(t) Use the check box to display the 95% CI for average difference. Report this interval.

Conclusions
(u) Write a paragraph summarizing your conclusion as if to another student in this class. Your conclusion should address (and justify) the statistical significance of the data, your estimate of the size of the parameter, whether you are willing to draw a cause-and-effect conclusion, and the population to which you are willing to generalize the results.
For this lab, after you submit Jotform will send you an email with a pdf attachment. Upload this pdf file into Canvas.

If you worked with a partner on this lab, both of you need to join a Investigation 8 group before submitting.
Should be Empty:

Background

Study Design

Paired Designs

Analyzing Paired Data

Analyzing Paired Data - Simulation

Analyzing Paired Data - Mathematical Model (Theory-Based)

Conclusions