91Ó°ÊÓ

The paper "Predictors of Complementary Therapy Use Among Asthma Patients: Results of a Primary Care Survey" (Health and Social Care in the Community [2008]: \(155-164)\) described a study in which each person in a large sample of asthma patients responded to two questions: Question 1: Do conventional asthma medications usually help your symptoms? Question 2: Do you use complementary therapies (such as herbs, acupuncture, aroma therapy) in the treatment of your asthma? Suppose that this sample is representative of asthma patients. Consider the following events: \(E=\) event that the patient uses complementary therapies \(F=\) event that the patient reports conventional medications usually help The data from the sample were used to estimate the following probabilities: $$ P(E)=0.146 \quad P(F)=0.879 \quad P(E \cap F)=0.122 $$ a. Use the given probability information to set up a hypothetical 1000 table with columns corresponding to \(E\) and not \(E\) and rows corresponding to \(F\) and not \(F\). b. Use the table from Part (a) to find the following probabilities: i. The probability that an asthma patient responds that conventional medications do not help and that the patient uses complementary therapies. ii. The probability that an asthma patient responds that conventional medications do not help and that the patient does not use complementary therapies. iii. The probability that an asthma patient responds that conventional medications usually help or the patient uses complementary therapies. c. Are the events \(E\) and \(F\) independent? Explain.

Step by step solution

01

Find the probabilities of each of the four categories in the 1000 table

First, we find the probabilities of the four categories in the 2x2 table: - P(E and F): Given as 0.122 - P(E and not F): P(E) - P(E and F) = 0.146 - 0.122 = 0.024 - P(not E and F): P(F) - P(E and F) = 0.879 - 0.122 = 0.757 - P(not E and not F): By subtraction since total probability is 1, 1 - P(E) - P(F) + P(E and F) = 1 - 0.146 - 0.879 + 0.122 = 0.097

02

Create the 1000 table

Using these probabilities, we can create a table with a hypothetical sample size of 1000 patients. Multiply each probability by 1000 to get the frequencies in each category. E not E Total F 122 757 879 not F 24 97 121 Total 146 854 1000 #b. Find probabilities using the table#

03

Calculate probabilities

Use the 1000 table to find the required probabilities: i. P(not F and E) = Frequency of (not F and E) / Total sample size = 24 / 1000 = 0.024 ii. P(not F and not E) = Frequency of (not F and not E) / Total sample size = 97 / 1000 = 0.097 iii. P(F or E) = 1 - P(not F and not E) = 1 - 0.097 = 0.903 #c. Check for independence#

04

Determine if E and F are independent

To determine if events E and F are independent, we can use the property that if E and F are independent, then P(E and F) = P(E) * P(F). If this property holds true, then the events are independent. If not, they are dependent. P(E and F) = 0.122 P(E) * P(F) = 0.146 * 0.879 = 0.12834 Since P(E and F) ≠ P(E) * P(F), the events E and F are not independent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is the branch of mathematics focused on analyzing random phenomena and quantifying the likelihood of different outcomes. It deals with the calculation of probabilities, the study of events, and outcomes within a given set of possible occurrences. In our exercise, probability theory is applied to evaluate the chance of asthma patients using complementary therapies and finding relief in conventional medications.

For instance, the probability of an event, represented as P(event), is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The exercise uses probability values to describe the likelihood of patients responding to therapies, as extracted from the given study. Calculations like P(E) = 0.146 tell us there's a 14.6% chance a patient will use complementary therapies. Understanding the basic principles of probability is essential for interpreting and solving such problems.

Statistical Independence

Statistical independence is a key concept in probability theory that describes a situation where the occurrence of one event does not affect the probability of the occurrence of another event. Two events are considered independent if and only if the probability of their intersection equals the product of their individual probabilities: P(A and B) = P(A) * P(B).

If we apply this concept to the exercise, we needed to check if the event of using complementary therapies (E) and the event of finding conventional medications helpful (F) are independent. Independence would imply that a patient’s experience with conventional medication has no bearing on their likelihood to use complementary therapies or vice versa. However, the calculation in the solution shows that the product of P(E) and P(F) does not equal P(E and F), indicating that these events are not independent. Consequently, the decision of a patient to use complementary therapies might be related to their experience with conventional medications.

Conditional Probability

Conditional probability focuses on the probability of an event occurring given that another event has already occurred. This concept is denoted as P(A|B), which reads as 'the probability of A given B.' It is calculated as P(A and B) / P(B) as long as P(B) is not zero.

In the context of our exercise, we might be interested in the conditional probability of a patient using complementary therapies given they don't find conventional medications helpful, P(E|not F). This wasn't directly calculated in the solution, but it is a natural extension of the concepts applied. If we had the number of people who don't find conventional meds helpful, we could calculate P(E|not F) using the information from the table. Understanding conditional probability is crucial for interpreting such relationships in data and making informed decisions based on existing conditions.