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Chapter 6: Problem 35

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. \({ }^{8}\) Assume the sample is reasonably representative of all such fourth down situations in the NFL.

Short Answer

Expert verified
The 95% confidence interval for the proportion of times NFL teams punt on a fourth down when, statistically speaking, they shouldn't punt, is calculated to be the range (lower limit, upper limit), where lower limit and upper limit are the results obtained in step 3.

Step by step solution

01

Identify the sample proportion and sample size

Identify the sample proportion (p) and the sample size (n). Here, the sample proportion is the proportion of times teams punted when they shouldn't have, given as 959 out of 1068. So, \(p = \frac{959}{1068}\). The sample size \(n = 1068\).
02

Calculate Standard Error

The standard error (SE) of a proportion is given by the formula \(\sqrt{p(1-p)/n}\), where p is the sample proportion and n is the sample size. Substitute the identified values of p and n.
03

Determine the 95% Confidence Interval

The 95% confidence interval is calculated using the formula \(p \pm Z * SE\), where Z is the z-value from the standard normal distribution corresponding to the desired confidence level. Here Z = 1.96 for a 95% confidence level. Substitute the calculated values of p, Z and SE into the formula to get the confidence interval.

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Key Concepts

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

Sample Proportion
The concept of a sample proportion is straightforward yet essential in statistical analysis. It's all about understanding parts of the whole.
For instance, if we look at a group of events and the number of times a specific event occurs, the sample proportion tells us how often we observe that event out of the total observations.
For our exercise, the economist found that NFL teams punted 959 times out of 1068 scenarios, when they shouldn't have, according to statistical reasoning.
To find the sample proportion, you simply divide the number of successful punts by the total number of fourth-down situations. That's calculating
  • Number of unwanted punts: 959
  • Total scenarios: 1068
  • Sample Proportion ( p ): \( p = \frac{959}{1068} \approx 0.897 \)
This value, 0.897, tells us that approximately 89.7% of the time, the teams decided to punt when statistics suggested otherwise.
Understanding the sample proportion helps us see the relative frequency of an event, and in this case, how often teams made questionable punting decisions.
Standard Error
Standard error is as important as it is intriguing. It gives us insight into the stability and reliability of a sample proportion by showing the expected variation from the true population proportion.
The formula used to determine the standard error for a proportion is: \[ SE = \sqrt{\frac{p(1-p)}{n}} \] where:
  • p is the sample proportion (0.897 in our case),
  • n is the sample size (1068).
Plugging in our numbers, we get: \[ SE = \sqrt{\frac{0.897 \times (1-0.897)}{1068}} \approx 0.0098 \]What does this mean?This value indicates the average distance that the sample proportion would lie from the true population proportion.
Essentially, it provides an understanding of how much we might expect our sample proportion to "bounce around" the actual proportion in the entire population.
The smaller the standard error, the more confidence we can have in our sample proportion being a good estimate of the real thing.
Statistical Analysis
Performing a statistical analysis, like calculating confidence intervals, helps us make informed decisions based on data.
In the context of our problem, we want to determine a range that estimates how often teams should wrongfully punt over the supports of endless games.
By using the confidence interval formula, we add and subtract a margin of error from our sample proportion:
\[ CI = p \pm Z * SE \] where:
  • p is the sample proportion (0.897),
  • Z is the z-score corresponding to the desired confidence level (1.96 for 95% confidence),
  • SE is the standard error (0.0098).
Substituting the values: \[ CI = 0.897 \pm 1.96 \times 0.0098 \] The result:This calculation yields an interval of \((0.878, 0.916)\).
In essence, we are 95% confident that the true proportion of punts, statistically classified as a bad decision, lies between 87.8% and 91.6%.
This statistical measure equips teams, referees, and analysts with a tool to question decisions and explore strategic improvements during games.

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Most popular questions from this chapter

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Exercise 4.86 on page 263 introduces a matched pairs study in which 47 participants had cell phones put on their ears and then had their brain glucose metabolism (a measure of brain activity) measured under two conditions: with one cell phone turned on for 50 minutes (the "on" condition) and with both cell phones off (the "off" condition). Brain glucose metabolism is measured in \(\mu \mathrm{mol} / 100 \mathrm{~g}\) per minute, and the differences of the metabolism rate in the on condition minus the metabolism rate in the off condition were computed for all participants. The mean of the differences was 2.4 with a standard deviation of \(6.3 .\) Find and interpret a \(95 \%\) confidence interval for the effect size of the cell phone waves on mean brain metabolism rate.

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