Great! Let's break down each part of the question step-by-step using the given joint relative frequencies from the two-way table provided.
### Part (a)
Question:
What is the probability that a randomly selected patient located in Saratoga was satisfied with the communication of the physician?
Solution:
To find the probability that a randomly selected patient located in Saratoga was satisfied with the communication (answered "Yes"), we directly look at the joint relative frequency for patients in Saratoga who were satisfied.
From the table:
- The relative frequency for satisfied patients in Saratoga (Yes) is [tex]\( 0.288 \)[/tex].
Thus, the probability that a randomly selected patient from Saratoga was satisfied with the communication is:
[tex]\[ \boxed{0.288} \][/tex]
### Part (b)
Question:
What is the probability that a randomly selected patient who was not satisfied with the physician's communication is located in Glens Falls?
Solution:
To determine this probability, we'll use conditional probability. We need to find the probability that a patient is from Glens Falls given that they were not satisfied.
1. Calculate the total probability of not being satisfied:
From the table:
- Glens Falls (No): [tex]\( 0.042 \)[/tex]
- Saratoga (No): [tex]\( 0.077 \)[/tex]
- Albany (No): [tex]\( 0.131 \)[/tex]
Total probability of a patient not being satisfied (No) is:
[tex]\[
0.042 + 0.077 + 0.131 = 0.25
\][/tex]
2. Calculate the conditional probability:
The probability that the not satisfied patient is from Glens Falls is the relative frequency of not satisfied patients from Glens Falls divided by the total probability of not satisfied patients.
Using the probability we found:
[tex]\[
\text{Probability} = \frac{\text{Probability of Glens Falls (No)}}{\text{Total Probability of Not Being Satisfied}}
\][/tex]
[tex]\[
\text{Probability} = \frac{0.042}{0.25} = 0.168
\][/tex]
Thus, the probability that a randomly selected patient who was not satisfied with the physician’s communication is located in Glens Falls is:
[tex]\[ \boxed{0.168} \][/tex]
In summary:
- Part (a): The probability that a randomly selected patient from Saratoga was satisfied with the physician's communication is [tex]\( 0.288 \)[/tex].
- Part (b): The probability that a randomly selected patient who was not satisfied with the physician's communication is from Glens Falls is [tex]\( 0.168 \)[/tex].
