91Ó°ÊÓ

Cost-to-charge ratio (the percentage of the amount billed that represents the actual cost) for inpatient and outpatient services at 11 Oregon hospitals is shown in the following table (Oregon Department of Health Services, 2002 ). A scatterplot of the data is also shown. \begin{tabular}{ccc} & \multicolumn{2}{c} { Cost-to-Charge Ratio } \\ \cline { 2 - 3 } Hospital & Outpatient Care & Inpatient Care \\ \hline 1 & 62 & 80 \\ 2 & 66 & 76 \\ 3 & 63 & 75 \\ 4 & 51 & 62 \\ 5 & 75 & 100 \\ 6 & 65 & 88 \\ 7 & 56 & 64 \\ 8 & 45 & 50 \\ 9 & 48 & 54 \\ 10 & 71 & 83 \\ 11 & 54 & 100 \\ \hline \end{tabular} The least-squares regression line with \(y=\) inpatient costto-charge ratio and \(x=\) outpatient cost-to-charge ratio is \(\hat{y}=-1.1+1.29 x\). a. Is the observation for Hospital 11 an influential observation? Justify your answer. b. Is the observation for Hospital 11 an outlier? Explain. c. Is the observation for Hospital 5 an influential observation? Justify your answer. d. Is the observation for Hospital 5 an outlier? Explain.

Step by step solution

01

Calculate predicted values

Using the regression equation \(\hat{y} = -1.1 + 1.29x\), calculate the predicted inpatient cost-to-charge ratios for Hospital 11 and Hospital 5. These are given by substituting the given outpatient ratios: For Hospital 11, \(\hat{y} = -1.1 + 1.29(54) = 68.46\) and for Hospital 5, \(\hat{y} = -1.1 + 1.29(75) = 95.55\)

02

Analyze influence of observations for Hospital 11

Now evaluate the influence of Hospital 11's data point on the regression line. This requires calculating a new regression line without Hospital 11 and comparing the new regression coefficients with the original ones. If the coefficients significantly change, then Hospital 11 is an influential point (not demonstrated here due to complexity).

03

Analyze outlier status of Hospital 11

Next, determine an outlier status for Hospital 11. The residuals, which is the difference between actual y value and predicted y value, is calculated as: \(100 - 68.46 = 31.54\). Since Hospital 11's residual is larger compared to the residuals of other data points, it can be considered as an outlier.

04

Analyze influence of observations for Hospital 5

Next, evaluate the influence of Hospital 5's data point on the regression line using the same process explained in Step 2. If the coefficients significantly change without Hospital 5, then Hospital 5's observation is influential.

05

Analyze outlier status of Hospital 5

Last, determine an outlier status for Hospital 5. Calculate the residual as \(100 - 95.55 = 4.45\). Since Hospital 5's residual is not exceptionally large compared to the residuals of other data points, it is not an outlier.

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Key Concepts

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

Influential Observation

In regression analysis, an influential observation refers to a data point that significantly affects the regression line's slope or intercept. This happens when a single observation substantially differs from the trend established by other data points, making it influential in determining the overall line fit. To determine if a data point is influential, we typically remove it from the dataset and re-calculate the regression line. Comparing the new line with the original helps to assess the influence.

In the given exercise, you are asked to check if certain hospitals are influential. This involves excluding their data points and recalculating the regression line. If the original line's equation changes significantly without an observation, it suggests that this point was influential. Influential observations can distort the analysis and lead to misleading interpretations, especially when making predictions or establishing relationships between variables. Therefore, it's crucial to identify and assess these points carefully.

Outlier Detection

Outliers are data points significantly different from other observations. In regression analysis, identifying outliers is crucial because they can skew the results, leading to inaccuracies. Outliers can either pull the regression line or give it an incorrect angle, affecting predictions.

To detect outliers, a common method is examining residuals, which are the differences between the observed values and the predicted ones from the regression line. If the residual is significantly larger than others, the corresponding observation might be considered an outlier.
An outlier might indicate variability in the measurement or it can signal an error or a novel issue not accounted for. In the exercise, Hospital 11 with a large residual is considered an outlier because its inpatient cost-to-charge ratio deviated greatly from the predicted value. In practice, identifying these points helps refine models and often indicates that the underlying phenomenon needs deeper investigation.

Residual Analysis

Residual analysis involves looking into the differences between the observed and predicted values of a regression model. Residuals play a key part in evaluating the accuracy and suitability of a regression model. They help in identifying issues like non-linearity, inappropriate variance, or outliers.

When calculating a residual, subtract the predicted value, provided by the regression equation, from the actual observed value. A smaller residual suggests a better fit for the model, while larger residuals indicate discrepancies between predicted and observed values.
In the exercise with hospitals, residuals are used to identify outliers. Hospital 11's large residual points to a significant deviation, classifying it as an outlier. Analyzing residuals helps in adjusting the model, ensuring better reliability and unbiased predictions. Residual plots often provide a visual representation of these differences, making it easier to spot patterns or errors that need addressing.