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In analyzing data from over 700 games in the National Football League, economist David Romer identified 1068 fourth-down situations in which, based on his analysis, the right call would have been to go for it and not to punt. Nonetheless, in 959 of those situations, the teams punted. Find and interpret a \(95 \%\) confidence interval for the proportion of times NFL teams punt on a fourth down when, statistically speaking, they shouldn't be punting. \(^{6}\) Assume the sample is reasonably representative of all such fourth down situations in the NFL.

Short Answer

Expert verified
The 95% confidence interval is (0.880, 0.916). This implies that between 88% and 91.6% of the time, NFL teams tend to punt when, based on the statistical analysis, they should not.

Step by step solution

01

Define the problem variables

In this case, the number of successes, i.e., the times the teams unexpectedly decided to punt, is 959. The total number of trials, representing all the fourth-down situations, is 1068. Also, know that the Z score for 95% confidence is 1.96.
02

Find the sample proportion

The sample proportion (p̂) is found by dividing the number of successes by the total number of trials. Hence, p̂ = 959 / 1068 = 0.898.
03

Calculate the standard error

The standard error (SE) = sqrt[ p̂*(1-p̂) / n ] = sqrt[ 0.898*(1-0.898) / 1068 ] = 0.009.
04

Determine the confidence interval

The general formula for a confidence interval for a proportion is p̂ ± Z * SE. Substituting in known values, the interval is therefore: 0.898 ± 1.96*0.009 ≈ (0.880, 0.916).
05

Interpret the confidence interval

Based in this 95% confidence interval, between 88% and 91.6% of the time, given fourth-down situations, NFL teams should not punt. This shows a strong likelihood of teams punting when, statistically speaking, they should be going for it.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Analysis
Understanding statistical analysis is crucial when evaluating actions and decisions in any data-driven context, including sports like the NFL. This process involves collecting and scrutinizing every piece of data to make informed conclusions. In the context of the exercise, the analysis focuses on determining the proportion of times NFL teams decided to punt on fourth down when statistics suggest they shouldn't have.

The study of these situations employs a statistical method known as a confidence interval, a type of estimate that gauges the precision of the sample proportion. Confidence intervals provide a range of values, which, with a certain level of confidence (95% in this case), is believed to capture the true population parameter. To construct a confidence interval, two main components are required: the sample proportion and the standard error. With these, analysts can extrapolate the data from the 1068 games sampled to make broader assertions about NFL decision-making habits.
NFL Decision Making
Decision-making in the NFL, especially on fourth downs, often intertwines with high-pressure situations and the complex strategies of the game. Coaches must weigh the risks and benefits of each available option, considering factors such as score, time remaining, and field position. Understanding the statistical implications of these decisions can be invaluable.

In the exercise, we see that empirical evidence from past games suggests a different approach to the conventional 'punt on fourth down' strategy. By using a rigorous statistical analysis, teams can potentially increase their chances of winning. However, NFL coaches might be influenced by tradition, risk aversion, or other non-statistical factors, which sometimes results in decision-making that deviates from what the statistical analysis recommends. Incorporating data-driven decisions can offer an edge, but it also requires a culture shift and acceptance of analytical insights in high-stakes environments.
Proportion Analysis
Proportion analysis is a fundamental aspect of statistical analysis, particularly when evaluating categorical data like the 'yes' or 'no' decision to punt the ball. It deals with the ratio of a particular outcome to the total number of observations or trials. In our example, the specific proportion is the number of times teams punted when they shouldn't have divided by the total number of situations examined.

Calculating the sample proportion (\( \text{p̂} \)) gives a snapshot of what happened in the observed cases, while computing the standard error (SE) accounts for variability and uncertainty in the sample. Both these statistics facilitate the calculation of the confidence interval. The interval assists in understanding not just the observed proportion but, more importantly, estimates the true proportion of such decisions if we could study the entire population of NFL fourth-down decisions. Proportion analysis thus underpins the broader field of inferential statistics, allowing analysts to make predictions and offer recommendations based on sampled data.

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