Task Sets: Decide if a model is consistent with data - Common Core Math - HS Statistics and Probability

Warm-Up: Models, Chance, and Distributions

These problems review ideas you already know about chance and distributions.

A fair coin is one that lands heads with probability $0.5$ . You flip a fair coin 5 times and get 5 tails in a row. Which statement is true?

This is an unlikely result, but it is possible and does happen for genuinely fair coins.

This is impossible for a fair coin, so the coin must be rigged.

This proves the coin is biased toward tails.

A fair coin can never produce a streak, so the result cannot occur.

A free-throw shooter claims she makes $80\%$ of her shots. In one session she takes 20 shots. You want to simulate this session to see how many makes are typical. What should one trial of your simulation produce?

The number of makes out of 20 simulated shots.

A single simulated shot: either "make" or "miss."

The probability $0.8$ .

The total number of makes across all 1000 repetitions.

A simulated distribution of the number of heads in 10 flips of a fair coin centers at its expected value. For a fair coin ( $p = 0.5$ ), the expected number of heads in 10 flips is $10 \times 0.5$ . About what value does the center of the simulated distribution sit at?

Fluency Practice

Apply the simulation framework: device, trial, statistic, repetitions, and reading distributions.

A model says a spinning coin lands heads with probability $0.5$ . Rank these results from LEAST surprising to MOST surprising under this model: getting 3 tails in a row, 5 tails in a row, 10 tails in a row.

3 tails (least), then 5 tails, then 10 tails (most surprising).

10 tails (least), then 5 tails, then 3 tails (most surprising).

All three are equally surprising, since each is possible.

5 tails (least), then 3 tails, then 10 tails (most surprising).

A spinner model claims red comes up with probability $0.3$ . To simulate using a random-digit table (digits $0$ – $9$ ), which assignment of digits correctly matches the model's probability for "red"?

Digits 0, 1, 2 = red; digits 3–9 = not red.

Digit 0 = red; digits 1–9 = not red.

Digits 0–4 = red; digits 5–9 = not red.

Any single digit = red, the rest = not red, since the device does not matter.

A simulation has these design choices. The model is "a die is fair." We want to know if getting four 6's in 12 rolls is surprising. Fill in each design choice: the chance device that matches the model is a fair ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ -sided die; one trial is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ simulated rolls; the statistic recorded is the number of ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ in those rolls.

number of sides on the die:

rolls per trial:

outcome being counted:

A simulated distribution of "number of heads in 10 flips of a fair coin" was built from 200 trials. The dot plot shows: most trials landed between 3 and 7 heads, the center is at 5, and exactly 2 of the 200 trials produced 9 or more heads. Estimate the fraction of trials that produced 9 or more heads (write as a decimal).

$Dot plot of simulated makes out of 20 under p = 0.8 across 50 trials; dots cluster near 16, and the observed value 11 is marked with a red dot far in the lower tail.$

The dot plot below is a simulated distribution of the number of makes in 20 free throws under the model $p = 0.8$ (50 simulated sessions). The shooter's actual session, 11 makes out of 20, is marked with the red dot far in the lower tail. Describe where the center of the distribution is, and explain in one sentence whether 11 makes looks typical or surprising under this model.

Exercises: Decide if a Model Is Consistent with Data Using Simulation

Warm-Up: Models, Chance, and Distributions

Fluency Practice