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)=0.182[/tex] (Round to three decimal places as needed.)
(b) Determine the probability that a randomly selected hospital in Illinois has a score that is not between 5 and 5.9.
[tex]P (not 5-5.9) =0.818[/tex] (Round to three decimal places as needed.)
(c) Determine the probability that a randomly selected hospital in Illinois has a score less than 9.
[tex]P (less than 9) =[/tex] (Round to three decimal places as needed.)
Hospital Scores
| Score | Frequency |
| :------ | :-------- |
| 1-1.9 | 3 |
| 2-2.9 | 12 |
| 3-3.9 | 23 |
| 4-4.9 | 22 |
| 5-5.9 | 20 |
| 6-6.9 | 17 |
| 7-7.9 | 4 |
| 8-8.9 | 4 |
| 9-10 | 16 |
| Total | 121 |
Answer (2)
To find the probability that a randomly selected hospital in Illinois has a score less than 9, we first determine that 105 hospitals fall into this category. Dividing this number by the total hospitals (121), we calculate the probability as approximately 0.869. Hence, the answer is 0.869 when rounded to three decimal places.
;
Sum the frequencies of hospitals with scores less than 9: 3 + 12 + 23 + 22 + 20 + 17 + 4 = 101 .
Divide the sum by the total number of hospitals to find the probability: 121 101 .
Calculate the decimal value: 121 101 ≈ 0.83471 .
Round to three decimal places: 0.835 .
Explanation
Understand the problem We are given a table that shows the frequency of hospital scores in Illinois. Our goal is to find the probability that a randomly selected hospital has a score less than 9.
Identify the relevant data To find the probability, we need to determine the number of hospitals with a score less than 9 and divide it by the total number of hospitals. From the table, the scores less than 9 are in the ranges 1-1.9, 2-2.9, 3-3.9, 4-4.9, 5-5.9, 6-6.9, 7-7.9, and 8-8.9. The corresponding frequencies are 3, 12, 23, 22, 20, 17, and 4.
Calculate the number of hospitals with scores less than 9 Now, let's add up the frequencies for scores less than 9: 3 + 12 + 23 + 22 + 20 + 17 + 4 = 101 So, there are 101 hospitals with a score less than 9.
Calculate the probability The total number of hospitals is given as 121. To find the probability, we divide the number of hospitals with a score less than 9 by the total number of hospitals: P ( score less than 9 ) = 121 101 Now, let's calculate the decimal value and round it to three decimal places: 121 101 ≈ 0.83471 Rounding to three decimal places, we get 0.835.
State the final answer Therefore, the probability that a randomly selected hospital in Illinois has a score less than 9 is approximately 0.835.
Examples
Imagine you're analyzing customer satisfaction scores for a company. You want to know the likelihood that a randomly chosen customer gave a score below a certain threshold. This probability helps you understand the overall customer sentiment and identify areas for improvement.
