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Chapter 12: Problem 8

Refer to Exercise \(12.45 .\) a. Estimate the average number of passing yards for games in which Brees throws 20 completed passes using a \(95 \%\) confidence interval. b. Predict the actual number of passing yards for games in which Brees throws 20 completed passes using a \(95 \%\) confidence interval. c. Would it be advisable to use the least-squares line from Exercise 12.45 to predict Brees' total number of passing yards for a game in which he threw only five completed passes? Explain.

Short Answer

Expert verified
In this exercise, we assume that we have the necessary data and statistical measures to estimate and predict the average number of passing yards using a 95% confidence interval for a given number of completed passes. We follow three steps and the main goal is to determine if it is advisable to use the least-squares line to predict Brees' total number of passing yards for a game in which he threw only five completed passes.

Step by step solution

01

Estimate the average number of passing yards for games in which Brees throws 20 completed passes using a 95% confidence interval

Given the slope, the intercept, and standard deviation of errors of the regression line and sample size from Exercise 12.45, we can estimate the average number of passing yards using the following interval formula: $$\overline{y} \pm t^* \frac{s_e}{\sqrt{n}}$$ Where: - \(\overline{y}\) is the predicted average number of passing yards for 20 completed passes - \(t^*\) is the t-distribution critical value for a 95% confidence interval - \(s_e\) is the standard deviation of errors - \(n\) is the sample size First, find the predicted average number of passing yards (\(\overline{y}\)) for 20 completed passes using the regression line equation: $$\overline{y} = a + b(20)$$ where - \(a\) is the y-intercept of the regression line - \(b\) is the slope of the regression line Next, compute the critical value \(t^*\) by finding the appropriate value from the t-distribution table using the degrees of freedom (\(n - 2\)) and a two-tailed 95% confidence level. Calculate the margin of error by multiplying the critical value by the fraction \(\frac{s_e}{\sqrt{n}}\). Finally, apply the interval formula to find the confidence interval for the average number of passing yards.
02

Predict the actual number of passing yards for games in which Brees throws 20 completed passes using a 95% confidence interval

To predict the actual number of passing yards, we use the prediction interval formula: $$\hat{y} \pm t^*\sqrt{s_e^2 + \frac{s_e^2}{n}}$$ Where: - \(\hat{y}\) is the predicted passing yards for a game in which Brees throws 20 completed passes - \(t^*\) is the t-distribution critical value for a 95% confidence interval - \(s_e\) is the standard deviation of errors - \(n\) is the sample size Note that both \(\hat{y}\) and \(\overline{y}\) are the same since they are both predictions for the same number of completed passes (20 completed passes). Compute the prediction interval by substituting the values and calculate the margin of error by multiplying the critical value by the square root of the sum of the squared standard deviation of errors and the squared standard deviation of errors divided by the sample size. Finally, apply the prediction interval formula to find the confidence interval for predicting the actual number of passing yards.
03

Determine if it's advisable to use the least-squares line to predict Brees' total number of passing yards for a game in which he threw only five completed passes

To determine if it's advisable to use the regression line to predict the total number of passing yards for a game in which Brees threw only five completed passes, consider the given data and the correlation coefficient. If the correlation coefficient is high and the data distribution is reasonable, it may be appropriate to use the regression line. Check if in the other data provided in the sample, there are games with few completed passes or if the data is concentrated only in a specific range of completed passes. If the correlation coefficient is low, or the data doesn't reasonably represent a game with five completed passes, then it may not be advisable to use the least-squares line for this prediction.

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Key Concepts

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

Least-Squares Regression
Least-squares regression is a method used to find the best-fit line through a set of data points on a graph. This line helps to predict the outcome of one variable based on another. In the context of our exercise, we use this line to predict the average number of passing yards Brees might have when he completes a certain number of passes, like 20 or 5 in one of the steps.

  • The least-squares regression line is determined by two key components: the slope and the intercept.
  • The slope indicates how much the passing yards are expected to change when Brees completes one more pass.
  • The intercept is the point where the line crosses the y-axis, showing expected passing yards when no passes are completed.
While using this line, it's crucial to check if the data fits the assumptions of linear regression like normality and homoscedasticity (constant variance) of errors. If the fit is not good, predictions might not be reliable.
T-Distribution
The t-distribution is a crucial concept when making predictions with a small sample size, as it accounts for extra uncertainty. In our exercise, it helps us determine the critical value, denoted as \(t^*\), necessary for confidence and prediction intervals. These intervals give us a range within which we expect the actual value to fall with a certain level of confidence, typically 95% as in our case.

  • This distribution is similar to the standard normal distribution but has fatter tails, allowing for more spread-out values, which is handy for small sample sizes.
  • The critical value \(t^*\) is determined from the t-distribution table, considering degrees of freedom \((n - 2)\), which accounts for the number of predictors we have used in the regression.
The t-distribution gets closer to the normal distribution as the sample size increases, meaning the predictions become more certain as more data is available.
Prediction Interval
A prediction interval gives a range within which we expect the actual data point to fall based on the regression model. It is different from a confidence interval which estimates where the average of many data points might be. In the context of our exercise, the prediction interval lets us predict the passing yards for a specific game where Brees completed 20 passes.

  • The formula for the prediction interval is \(\hat{y} \pm t^*\sqrt{s_e^2 + \frac{s_e^2}{n}}\), where \(\hat{y}\) is the predicted value.
  • This interval is wider than the confidence interval because it includes additional uncertainty due to the prediction of a single observation, not a mean.
By interpreting the prediction interval, coaches and analysts can have a range of expected performance, making game planning and decision-making more informed.
Statistical Estimation
Statistical estimation involves estimating population parameters based on sample data. In the context of our problem, we use statistical estimation to predict passing yards based on limited data about passes completed. This technique is fundamental in developing insights into a dataset and making educated predictions.

  • It uses existing sample data to infer about a larger population, making the analysis widely applicable to real-world scenarios.
  • Statistical estimation is key in forming confidence intervals, which provide a range that is likely to contain the true population parameter.
Proper understanding of statistical estimation helps one make informed predictions using a regression line, by contextualizing them within a bigger picture beyond the sample. It turns raw data into actionable insights through methods like regression and hypothesis testing.

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Most popular questions from this chapter

An agricultural experimenter, investigating the effect of the amount of nitrogen \(x\) applied in 100 pounds per acre on the yield of oats \(y\) measured in bushels per acre, collected the following data: $$\begin{array}{l|lllll} x & 1 & 2 & 3 & 4 \\ \hline y & 22 & 38 & 57 & 68 \\\ & 19 & 41 & 54 & 65 \end{array} $$ a. Find the least-squares line for the data. b. Construct the ANOVA table. c. Is there sufficient evidence to indicate that the yield of oats is linearly related to the amount of nitrogen applied? Use \(\alpha=.05 .\) d. Predict the expected yield of oats with \(95 \%\) confidence if 250 pounds of nitrogen per acre are applied. e. Estimate the average increase in yield for an increase of 100 pounds of nitrogen per acre with \(99 \%\) confidence f. Calculate \(r^{2}\) and explain its significance in terms of predicting \(y,\) the yield of oats.

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Graph the line corresponding to the equation \(y=2 x+1\) by graphing the points corresponding to \(x=0,1,\) and 2 . Give the \(y\) -intercept and slope for the line.

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