In an effort to reduce the number of hospital-acquired conditions (such as infection resulting from the hospital stay), Medicare officials score hospitals on a 10-point scale with a lower score representing a better patient track record. The federal government reduces Medicare payments to those hospitals with the worst scores. The following data represent the scores received by a set of Illinois hospitals. Complete parts (a) through (d).
(a) Determine the probability that a randomly selected hospital in Illinois has a score between 5 and 5.9.
[tex]P(5.5.9)=\square \text { (Round to three decimal places as needed.) }[/tex]
Hospital Scores
| Score | Frequency |
| :------ | :-------- |
| 1-1.9 | 3 |
| 2-2.9 | 12 |
| 3-3.9 | 16 |
| 4-4.9 | 23 |
| 5-5.9 | 22 |
| 6-6.9 | 20 |
| 7-7.9 | 17 |
| 8-8.9 | 4 |
| 9-10 | 4 |
| Total | 121 |
Answer (2)
The probability that a randomly selected hospital in Illinois has a score between 5 and 5.9 is approximately 0.182. This is calculated by dividing the number of hospitals in this score range (22) by the total number of hospitals (121). After simplification, the probability is rounded to three decimal places as needed.
;
Identify the frequency of hospitals with scores between 5 and 5.9: 22.
Divide the frequency by the total number of hospitals to find the probability: 121 22 .
Simplify the fraction: 121 22 = 11 2 .
Round to three decimal places: 11 2 ≈ 0.182 .
Explanation
Understand the problem We are given a frequency table of hospital scores and asked to find the probability that a randomly selected hospital has a score between 5 and 5.9.
Identify the frequency and total number of hospitals From the table, we see that the frequency of hospitals with scores between 5 and 5.9 is 22. The total number of hospitals is 121.
Calculate the probability To find the probability, we divide the frequency of hospitals with scores between 5 and 5.9 by the total number of hospitals: P ( 5 l escore l e 5.9 ) = 121 22
Simplify the fraction Now, we simplify the fraction: 121 22 = 11 × 11 2 × 11 = 11 2
Round to three decimal places To round the fraction to three decimal places, we perform the division: 11 2 ≈ 0.181818... Rounding to three decimal places, we get 0.182.
State the final answer Therefore, the probability that a randomly selected hospital in Illinois has a score between 5 and 5.9 is approximately 0.182.
Examples
In quality control, this type of probability calculation can help determine the likelihood of a product falling within a certain acceptable range. For example, if you're manufacturing bolts and want to ensure they meet specific diameter requirements, you can measure a sample of bolts and calculate the probability that a randomly selected bolt falls within the desired diameter range. This helps you assess the consistency and quality of your manufacturing process. If a company wants to know the probability that their product will be of certain quality, they can use this method.
