[NW] Medicare Fines 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 Illinois hospitals.
c. Determine the probability that a randomly selected hospital in Illinois has a score less than 9.

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the probability that a randomly selected hospital has a score less than 9. The scores are grouped into intervals with corresponding frequencies.
2. Step 2: Identify the score intervals that are less than 9. These intervals are 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.
3. Step 3: Sum the frequencies of all these intervals to find the total number of hospitals with scores less than 9. This means adding the frequencies: 3 + 12 + 16 + 23 + 23 + 21 + 17 + 5.
4. Step 4: Use the total number of hospitals (125) as the denominator to calculate the probability. The formula for probability is \(P(\text{score} < 9) = \frac{\text{Number of hospitals with score} < 9}{\text{Total number of hospitals}}\).
5. Step 5: Write the probability as a fraction or decimal by dividing the sum of frequencies from Step 3 by 125. This gives the probability that a randomly selected hospital has a score less than 9.

Key Concepts

Frequency Distribution

A frequency distribution organizes data into intervals or categories and shows how many data points fall into each. It helps summarize large data sets and is essential for calculating probabilities based on observed frequencies.

Probability from Frequency Data

Probability can be estimated by dividing the frequency of the event of interest by the total number of observations. For example, the probability that a hospital score is less than 9 is the sum of frequencies for scores below 9 divided by the total number of hospitals.

Cumulative Frequency

Cumulative frequency is the running total of frequencies up to a certain class or score. It is useful for determining the number of observations below a specific value, which aids in calculating probabilities for ranges of data.