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Let's break down each part of the problem:

### Part (a): Longest Streak
To determine Roethlisberger's longest streak of completions ("Y"), let's count the consecutive "Y"s in the sequence. 

The sequence provided is:
```
Y N Y N N Y Y Y N Y Y N Y N N Y Y Y Y Y N Y N Y N Y Y Y Y Y Y N Y Y Y Y Y N N
```

If we look at the longest sequence of consecutive "Y"s:
- The longest streak is **6** consecutive "Y"s.

So, the answer for part (a) is:
**Roethlisberger’s longest streak during the game was 6 completions.**

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### Part (b): Hypotheses for Testing
In this context, the hypotheses would likely involve testing whether Roethlisberger's completions and incompletions occur randomly or if there is a tendency for streaks, indicating "streaky" performance. Here’s how we could frame the hypotheses:

- **Null Hypothesis (H₀):** Roethlisberger's pass completions and incompletions are random and follow a binomial distribution, indicating no streakiness in his passing.
- **Alternative Hypothesis (H₁):** Roethlisberger’s pass completions and incompletions are not random and exhibit streakiness, meaning he has clusters of successful or unsuccessful passes.

---

### Part (c): Meaning of "Streaky Passer"
To claim that Roethlisberger was a "streaky passer" means that he tends to go on runs of consecutive completions or failures, rather than each pass attempt being independent of previous attempts. In other words, his success in passing has a pattern or sequence rather than following a completely random distribution.

Would you like further details or explanations?

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#### Related Questions

1. How could we statistically test if Roethlisberger’s passing sequence is random?
2. What other factors could influence whether a player appears "streaky" in their performance?
3. How would increasing the number of attempts affect the likelihood of seeing longer streaks?
4. What is the difference between a "hot hand" and random clustering in statistical terms?
5. Could external conditions (like weather or defense quality) affect streakiness?

#### Tip
To identify streaks in any sequence, count consecutive occurrences of the same value, then compare across the entire sequence.

In the 2011 Super Bowl, quarterback Ben Roethlisberger of the Pittsburgh Steelers completed 25 of his 40 pass attempts. Here are the outcomes of his pass attempts, in chronological order: Y N Y N N Y Y Y N Y Y N Y N N Y Y Y Y Y N Y N Y N Y Y Y Y Y Y N Y Y Y Y Y N N.

(a) What was Roethlisberger’s longest streak during the game?

(b) State the hypotheses you are interested in testing.

(c) Briefly explain what it means to claim that Roethlisberger was a streaky passer.

Explore an analysis of Ben Roethlisberger’s 2011 Super Bowl passing sequence to determine his longest streak, form hypotheses for testing randomness, and understand what it means to call a passer 'streaky.' This analysis involves probability concepts and hypothesis testing, suitable for Grades 11-12.

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