91Ó°ÊÓ

Cost-to-charge ratios (see Example \(4.9\) for a definition of this ratio) were reported for the 10 hospitals in California with the lowest ratios (San Luis Obispo Tribune, December 15,2002 ). These ratios represent the 10 hospitals with the highest markup, because for these hospitals, the actual cost was only a small percentage of the amount billed. The 10 cost-to-charge values (percentages) were $$ \begin{array}{rrrrrr} 8.81 & 10.26 & 10.20 & 12.66 & 12.86 & 12.96 \\ 13.04 & 13.14 & 14.70 & 14.84 & & \end{array} $$ a. Compute the variance and standard deviation for this data set. b. If cost-to-charge data were available for all hospitals in California, would the standard deviation of this data set be larger or smaller than the standard deviation computed in Part (a) for the 10 hospitals with the lowest cost-to-charge values? Explain. c. Explain why it would not be reasonable to use the data from the sample of 10 hospitals in Part (a) to draw conclusions about the population of all hospitals in California.

Step by step solution

01

Calculate the Mean

Add all cost-to-charge values and divide by the number of values (10) to find the mean. The mean of the data set is \( \bar{x} = \frac{{8.81 + 10.26 + 10.20 + 12.66 + 12.86 + 12.96 + 13.04 + 13.14 + 14.70 + 14.84}}{10} = 12.357 \)

02

Find the Deviations

Compute the deviation of each data value from the mean. Deviation is the difference between each data value and the mean. For each data value, subtract the mean of 12.357.

03

Square the Deviations

Square each deviation computed in Step 2 to get the set of squared deviations.

04

Compute the Variance

Calculate the variance by adding all squared deviations computed in step 3 and divide them by the number of values. The variance, denoted by \( S^2 \), is \( S^2 = \frac{{\sum_{i=1}^{n} (x_i - \bar{x})^2}}{n-1} \) where \(n\) is the number of data value, \(x_i\) each data value and \(\bar{x}\) is the mean.

05

Compute the Standard Deviation

Compute the standard deviation by taking the square root of the variance computed in step 4. The standard deviation, denoted by \( S \), is \( S = \sqrt{S^2} \)

06

Analyze for Part B

In order to determine whether or not the standard deviation of cost-to-charge ratios for all hospitals in California would be larger or smaller than the standard deviation for the 10 hospitals with lowest cost-to-charge ratios, we need to understand the nature of the data. Because the data we have includes extremes (lowest cost-to-charge ratios), including data from all hospitals will most likely increase the range (highest minus lowest value), leading to a higher standard deviation.

07

Formulate Explanation for Part C

The sample of 10 hospitals, which were selected for having the lowest cost-to-charge ratios, may not be representative of all the hospitals in California. The data is skewed towards hospitals with low cost-to-charge ratios and does not take into consideration hospitals with higher cost-to-charge ratios. Using this sample to draw conclusion about all hospitals in California could likely result in biased or misleading findings.

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Key Concepts

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

Cost-to-Charge Ratios

Understanding the concept of cost-to-charge ratios is crucial in healthcare finance. These ratios represent the relationship between the cost incurred by a hospital to deliver services and the charges billed to the patients. Essentially, it measures how much more a hospital charges compared to the actual costs of service delivery. Typically, a lower cost-to-charge ratio indicates a higher markup. This is because the billed amount significantly surpasses the actual costs involved. Therefore, this is critical for patients, insurers, and policy-makers to understand how hospitals set their rates, and it is a vital tool for financial analysis in the healthcare industry.

Standard Deviation

Standard deviation is a key statistical tool used to measure the amount of variation or dispersion in a set of data values.
It tells us how spread out the data points are from the mean and provides insight into the consistency of data.
To calculate the standard deviation:

• First, find the variance by calculating the average of squared deviations from the mean.
• Then, take the square root of the variance.

For example, in the case of cost-to-charge ratios, a smaller standard deviation would indicate that the values are close to the mean, reflecting consistency among the hospitals in terms of their cost-to-charge practices. Conversely, a larger standard deviation would suggest more variability among the hospitals' pricing strategies, indicating less predictability.

Variance

Variance is another fundamental statistical concept, representing the degree of spread in a data set.
It gives the average of the squared differences from the mean, helping quantify the data's dispersion.
In essence, variance serves as a measure of how each number in a data set differs from the mean and thus from every other number in the set.

• To find the variance, we calculate the mean of the data set.
• Then, subtract the mean from each data point and square the result.
• Finally, find the average of these squared differences.

Because variance uses the square of deviations, it is expressed in square units. When considering financial and statistical analysis, like cost-to-charge ratios, recognizing variance helps in understanding how dispersed the data points are, which is critical for making informed decisions.

Sample Representativeness

Sample representativeness is highly important when drawing conclusions from a statistical analysis.
It reflects how well a sample reflects the entire population from which it is drawn.
A sample is considered representative if it equally represents the various segments of the entire population, without bias.

• If a sample is not representative, any conclusions drawn may be misleading.
• In the case of cost-to-charge ratios from only ten hospitals, the sample specifically highlights those with low ratios.
• This can lead to skewed results, as it doesn’t account for hospitals with higher ratios, thus not accurately representing the complete spectrum of hospital billing practices in California.

Thus, it's crucial to examine how a sample is chosen to ensure that it can reliably inform about the overall population.