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Chapter 3: Problem 14

For each of the following exercises, determine the range (possible values) of the random variable. A healthcare provider schedules 30 minutes for each patient鈥檚 visit, but some visits require extra time. The random variable is the number of patients treated in an eight-hour day.

Short Answer

Expert verified
The range of the random variable is 1 to 16 patients.

Step by step solution

01

Determine Total Available Time

First, calculate the total number of minutes available in an 8-hour day. Since there are 60 minutes in an hour, an 8-hour day has \( 8 \times 60 = 480 \) minutes available.
02

Determine Minimum Number of Patients

Consider the situation when each patient takes the maximum scheduled time of 30 minutes, meaning no extra time is needed. The maximum number of patients that can be treated is \( \frac{480}{30} = 16 \). Thus, the minimum value of the random variable is 0 patients (if the doctor does not see any patients for some reason), and in practical scenarios, it's important to consider that a minimum of 1 patient would be seen.
03

Determine Maximum Number of Patients

Assume each patient's visit takes less than 30 minutes, allowing more patients to be seen. Theoretical maximum happens if no time was consumed by any individual, which is unrealistic. Practically, if visits are shorter but each still uses some time, the most ideal realistic scenario could allow seeing up to 16 patients, considering some overlap and efficiency in scheduling.
04

Establish Range of Random Variable

Based on our calculations, the minimum realistic number of patients treated is 1, and the maximum number is 16. Therefore, the range of the random variable is from 1 to 16 patients.

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Key Concepts

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

Understanding Probability in Random Variables
Probability is a fundamental concept in mathematics that measures the likelihood of an event occurring. When dealing with random variables in exercises like determining patient visits, probability estimates how likely a particular outcome is.
For example, when considering how many patients a healthcare provider might see in an eight-hour day, probability helps us understand which numbers of patients are more likely.
  • If each patient's visit is exactly 30 minutes, it's highly probable to see close to 16 patients.
  • However, if visits vary in length, the probability of seeing anywhere from a few to 16 patients can fluctuate.
Understanding probability can give insight into how likely each possible outcome is, informing decisions and planning.
Exploring the Range of Values
The range of values is critical when analyzing a random variable since it defines the possible outcomes. In our case, the range is the number of patients that can be seen in a day.
To establish this, first, we calculate the maximum scheduled number of patients using the standard 30-minute time frame. Here, 480 minutes divided by 30 gives 16 patients.
  • Minimum Value: While theoretically starting at zero, practically it's unlikely to treat none, hence the range begins at 1.
  • Maximum Value: Under ideal conditions factoring quick turnover or efficiency, the upper limit remains realistic at 16.
Understanding this helps set realistic expectations and planning within healthcare scheduling.
Implementing Statistical Analysis
Statistical analysis allows for deeper exploration of data from random variables. When understanding how many patients can be seen, statistical tools can be applied to analyze trends or variations.
For example, if historical data shows patterns in patients being treated quicker due to streamlined processes or shorter consultations, adjustments to the expected number could be made.
Some key elements of statistical analysis include:
  • Mean and Median: The average number of patients typically seen versus the middle value of visits across data sets.
  • Variance and Standard Deviation: Measuring fluctuations from the average number of patients, helps assess day-to-day variability in visits.
Through statistical analysis, better models are developed for scheduling, enhancing both patient care efficiency and resource management.

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Determine the cumulative distribution function of a binomial random variable with \(n=3\) and \(p=1 / 4\).

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