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Statistics 9-12 / Week 09 / Monday
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Week 09 · Comparing Two Treatments

Monday

Two warm-ups and a shuffled deck
// Two warm-ups, one shuffled deck
⏱ about 20 min

Monday: Two Warm-Ups and a Shuffled Deck

"Which warm-up is better, the stretch routine or the loop jog?" Comet asks. "12 runners want to know."

"Then we do not let them choose," Wren says, shuffling a deck of six red and six blue cards.

"Red stretches. Blue jogs the loop. The card decides, not the runner."

Each runner draws a card, warms up, and runs one timed lap. Nova logs every second.

"Stretch group mean 94 seconds," Nova reports. "Jog group mean 90. Would you like a hint? Look at the difference."

"4 seconds faster after the jog," Comet says. "So the jog is better!"

"Maybe," Wren says. "What do you notice? Six runners each. Chance could have put the fast runners in blue."

"Then we need to know what chance alone can do," Comet says. "And this deck is going to tell us."

Two groups of runners stretch on the grass, Wren shuffles cards at a picnic table, Comet times a lap.

A randomized experiment

An experiment assigns a treatment. The crew's treatments are the stretch routine and the loop jog.

Random assignment lets chance choose the groups. With a shuffled deck, no runner, fast or slow, is more likely to land in either group.

That is why a randomized experiment can point to a cause: the only things separating the groups are the treatment and chance.

Every lap time in this course is the crew's own made-up reading from Nova's Run Log.

A solved problem to study

Here is how Wren finds the difference the crew observed.

  1. Stretch group lap times: 95, 92, 97, 94, 96, 90 seconds. Add them: 564. Divide by 6: mean 94.
  2. Jog group lap times: 88, 91, 86, 93, 90, 92 seconds. Add them: 540. Divide by 6: mean 90.
  3. Difference of means, stretch minus jog: 94 - 90 = 4 seconds.
  4. A positive difference means the jog group ran faster. The question for the week: could chance alone make a gap of 4?
GroupLap times (seconds)Mean
Stretch (red cards)95, 92, 97, 94, 96, 9094
Jog (blue cards)88, 91, 86, 93, 90, 9290
Two box plots of lap times: the stretch group sits around 94 seconds, the jog group around 90.
THE CREW'S EXPERIMENT
  • Read the question.
  • Tap your answer.
Stretch group: 95, 92, 97, 94, 96, 90 seconds. Jog group: 88, 91, 86, 93, 90, 92 seconds. What is the difference of the means, stretch minus jog? (Round to 1 place.)
Stretch group: 95, 92, 97, 94, 96, 90. Jog group: 88, 91, 86, 93, 90, 92. Which group has the larger mean lap time?
The crew used a shuffled deck to decide which warm-up each runner did, then timed a lap. What kind of study is this?
WHY SHUFFLE?
  • Read the question.
  • Tap your answer.
How does randomization work in an experiment?
Why did the crew let a card choose each runner's warm-up?
How many runners took part in the crew's experiment? Type the number.
WHY THIS EXERCISESmall groups make chance a bigger player. The crew will see exactly how big this week.
StatementTrue or false?
94 - 90 = 4?
Random assignment means the runner picks the warm-up they prefer.?
A difference of 4 seconds proves the jog is better.?
In a randomized experiment, only the treatment and chance separate the groups.?
WHY THIS EXERCISEThe whole week is one question: is this gap more than chance could make?
Try it
Write the 12 lap times on 12 index cards, one per card. Keep them for Wednesday's Data Lab.
Shuffle and deal six and six. Find each hand's mean and the difference. Was it bigger or smaller than 4?
Draw two dot plots of lap times, stretch above jog, on one number line from 85 to 98.

A strong start. Tomorrow the deck answers the question: what can chance alone do to twelve lap times?