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Let's work through the problem step by step. The question asks whether the New York Giants had a home-field advantage in the 2016 season based on their win-loss record at home and away.

### Step 1: Summarize the Data in a Two-Way Table
The New York Giants' performance is:
- Home: 7 wins, 1 loss
- Away: 4 wins, 5 losses

We can organize this data into a two-way table:

|            | Home | Away | Total |
|------------|------|------|-------|
| Wins       |   7  |   4  |   11  |
| Losses     |   1  |   5  |   6   |
| Total      |   8  |   9  |   17  |

This table provides a clear view of how many games they won and lost both at home and away.

### Step 2: Hypotheses
The next step is to define the null and alternative hypotheses:
- **Null Hypothesis (H₀):** There is no home-field advantage, meaning the Giants are equally likely to win whether they play at home or away.
- **Alternative Hypothesis (Hₐ):** There is a home-field advantage, meaning the Giants are more likely to win at home than away.

### Step 3: Calculate the Test Statistic
To determine whether the difference between home and away win percentages is statistically significant, we can use a proportion test. The formula for the test statistic for two proportions can be written as:

\[
z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{p(1-p) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}
\]

Where:
- \( \hat{p}_1 = \) proportion of wins at home = \( \frac{7}{8} \)
- \( \hat{p}_2 = \) proportion of wins away = \( \frac{4}{9} \)
- \( p \) is the pooled proportion of wins: \( p = \frac{7 + 4}{8 + 9} = \frac{11}{17} \)

Substitute the values and calculate the test statistic.

### Step 4: Index Card Simulation
To conduct an index card simulation, you would:
1. Write "Win" on 11 index cards (since the Giants won 11 games).
2. Write "Loss" on 6 index cards (since they lost 6 games).
3. Shuffle the cards and randomly assign them into two groups: one group of 8 cards (for home games) and one group of 9 cards (for away games).
4. Count how many wins and losses occur in each group.
5. Repeat this process many times (usually 1,000 or more) to simulate random outcomes.

### Step 5: Analyze Simulation Results
After performing the simulation, you would plot the differences in win percentages between the simulated home and away records on a dot plot. Then, compare the actual win percentage difference to this distribution to see if the observed difference could be due to random chance.

### Conclusion
If the actual difference between home and away win percentages falls far in the tail of the simulated distribution, we would reject the null hypothesis and conclude that the Giants had a statistically significant home-field advantage. If not, we fail to reject the null hypothesis and conclude that the difference could be due to random chance.

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Would you like to go into more detail on any of these steps or need further clarification?

### Related Questions:
1. How do you calculate the pooled proportion in a two-proportion z-test?
2. What is the purpose of performing a simulation in hypothesis testing?
3. How do we interpret the result of a z-test in terms of statistical significance?
4. Can a team have a statistically significant home advantage with a small sample size of games?
5. How could the dot plot be used to visualize the results of the simulation?

#### Tip: Always check if your sample size is large enough to apply normal approximation methods like the z-test for proportions. If the sample size is small, alternative methods may be needed!

Did the New York Giants have a home-field advantage? In the 2016 season, the New York Giants won 7 games and lost 1 at home, while winning only 4 games and losing 5 on the road. Is this difference statistically significant, or could this difference be attributed to random chance?

This problem explores whether the New York Giants had a home-field advantage in the 2016 season using hypothesis testing. Data includes a 7-1 home record and 4-5 away record. Learn to calculate the test statistic and run a simulation to check statistical significance.

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