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).
Click the icon to view the hospital scores.
(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.)
Hospital Scores
(b) Determine the probability that a randomly selected hospital in Illinois has a score that is not between 5
[tex]P ( not 5-5.9 )=[/tex] (Round to three decimal places as needed.)
| 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 that is not between 5 and 5.9 is 0.818. This is determined by subtracting the given probability of 0.182 from 1. Thus, the calculation shows a high chance that a hospital score falls outside of this range.
;
Calculate the probability of a hospital having a score between 5 and 5.9: P ( 5 − 5.9 ) = 0.182 .
Calculate the probability of a hospital NOT having a score between 5 and 5.9: P ( not 5 − 5.9 ) = 1 − 0.182 .
Subtract to find the complement: 1 − 0.182 = 0.818 .
The probability that a randomly selected hospital in Illinois has a score that is not between 5 and 5.9 is 0.818 .
Explanation
Understand the problem and provided data Let's analyze the problem. We are given the probability that a randomly selected hospital in Illinois has a score between 5 and 5.9, which is 0.182. We need to find the probability that a randomly selected hospital does NOT have a score between 5 and 5.9. This is the complement of the given probability.
Calculate the complement probability To find the probability of the complement, we subtract the given probability from 1. So, we have: P ( not 5 − 5.9 ) = 1 − P ( 5 − 5.9 ) P ( not 5 − 5.9 ) = 1 − 0.182
State the final answer Performing the subtraction, we get: P ( not 5 − 5.9 ) = 0.818 So, the probability that a randomly selected hospital in Illinois has a score that is not between 5 and 5.9 is 0.818.
Examples
Imagine you're tracking customer satisfaction scores for a product on a scale of 1 to 10. If you know the probability that a customer gives a score between 5 and 5.9 is 0.182, you can calculate the probability that a customer gives a score outside this range (either lower or higher) by subtracting 0.182 from 1. This helps you understand the overall distribution of satisfaction scores and identify areas for improvement.
