Chapter 4.2, Problem 1AC

Applying the Concepts 4–2

Which Pain Reliever Is Best?

Assume that following an injury you received from playing your favorite sport, you obtain and read information on new pain medications. In that information you read of a study that was conducted to test the side effects of two new pain medications. Use the following table to answer the questions and decide which, if any, of the two new pain medications you will use.

1. How many subjects were in the study?

2. How long was the study?

3. What were the variables under study?

4. What type of variables are they, and what level of measurement are they on?

5. Are the numbers in the table exact figures?

6. What is the probability that a randomly selected person was receiving a placebo?

7. What is the probability that a person was receiving a placebo or drug A? Are these mutually exclusive events? What is the complement to this event?

8. What is the probability that a randomly selected person was receiving a placebo or experienced a neurological headache?

9. What is the probability that a randomly selected person was not receiving a placebo or experienced a sinus headache?

To determine

To obtain: The number of objects in the study.

Answer to Problem 1AC

 The number of objects in the study is 566.

Explanation of Solution

Given info:

The data set represents the three groups. Those are placebo, Drug A and drug B.

Calculation:

The placebo group contains 192 subjects, group A contains 186 subjects and Group B contains 188 subjects.

The total number of subjects in the given study is $566\left(=192+186+188\right)$

Thus, the total number of subjects in the given study is 566.

To determine

To identify: The time duration for the study.

Answer to Problem 1AC

The time duration for the study is 12 weeks.

Explanation of Solution

Justification:

In a clinical trial, the number of side effects for three groups was observed for 12 weeks.

Thus, the time duration for the study is 12 weeks.

To determine

To identify: The variables in the study.

Answer to Problem 1AC

The variables in the study are type of pain reliever and the side effects.

Explanation of Solution

Justification:

In the study, the side effects for three groups are observed. That is, the study represents the number of side effects for each group.

Thus, the variables in the study are side effects and pain relievers.

To determine

To identify: The type of variables and the level of measurement.

Answer to Problem 1AC

The type of variable is qualitative and the level of measurement is nominal.

Explanation of Solution

Justification:

The variable ‘side effect’ represents 6 categories and ‘pain reliever’ represents 3 groups like placebo, group A and group B. Therefore, the variables in the study represent the qualitative variables. Moreover, the qualitative variables measured on nominal scale.

Thus, the type of variables is qualitative and the level of measurement is nominal.

To determine

To observe: Whether the figures in the table or exact or not.

Answer to Problem 1AC

The numbers in the study are exact figures.

Explanation of Solution

Justification:

The numbers in the table represents the number of side effects. Hence, the numbers are exact.

Thus, the numbers in the study are exact figures.

To determine

To obtain: The probability of randomly selected person was receiving a placebo.

Answer to Problem 1AC

The probability of randomly selected person was receiving a placebo is 0.339.

Explanation of Solution

Given info:

The table shows the number of side effects in placebo group is 192 and the total number of side effects is 566.

Calculation:

The formula for probability of randomly selected person with placebo is,

$P\left(\text{Receiving placebo}\right)=\frac{\text{Number of side effects in placebo}}{\text{Total number of side effects}}$

Substitute 192 for ‘Number of side effects in placebo’ and 566 for ‘Total number of side effects’,

$\begin{array}{c}P\left(\text{Receiving placebo}\right)=\frac{\text{192}}{\text{566}}\\ =0.339\end{array}$

Thus, the probability of randomly selected person was receiving a placebo is 0.339.

To determine

To obtain: The probability of randomly selected person was receiving a placebo or drug A.

To observe: Whether the events are mutually exclusive or not.

To identify: The complement to the event.

Answer to Problem 1AC

The probability of randomly selected person was receiving a placebo or drug A is 0.668.

The events placebo and drug A are mutually exclusive.

The complement for the event is the randomly selected person was receiving drug B.

Explanation of Solution

Given info:

The table shows the number of side effects in placebo group is 192, the number side effects in drug A is 186 and the total number of side effects is 566.

Calculation:

The formula for probability of randomly selected person with drug A is,

$P\left(\text{Receiving drug A}\right)=\frac{\text{Number of side effects in drug A}}{\text{Total number of side effects}}$

Substitute 186 for ‘Number of side effects in drug A’ and 566 for ‘Total number of side effects’,

$\begin{array}{c}P\left(\text{Receiving drug A}\right)=\frac{\text{186}}{\text{566}}\\ =0.329\end{array}$

Addition Rule:

The formula for probability of getting placebo group or drug A is,

$P\left(\text{placebo or drug A}\right)=P\left(placebo\right)+P\left(\text{drug }A\right)$.

Substitute 0.339 for ‘ $P\left(placebo\right)$’, 0.329 for ‘ $P\left(\text{drug A}\right)$’

$\begin{array}{c}P\left(\text{placebo or drug A}\right)=0.339+0.329\\ =0.668\end{array}$

Thus, the probability of randomly selected person was receiving a placebo or drug A is 0.668.

Justification:

There is intersection part between place and drug A. Hence, the placebo and drug A are mutually exclusive events. The complement event for ‘person receiving the placebo or drug A’ is the randomly selected person was receiving drug B.

Thus, the events placebo and drug A are mutually exclusive and the complement for the event is the randomly selected person was receiving drug B.

To determine

To obtain: The probability of randomly selected person was receiving a placebo or experienced a neurological headache.

Answer to Problem 1AC

The probability of randomly selected person was receiving a placebo or experienced a neurological headache is 0.564.

Explanation of Solution

Given info:

The table shows the number of side effects in placebo group is 192, the number side effects in placebo and neurological headache is 34.

Calculation:

The formula for probability of randomly selected person with neurological headache is,

$P\left(\text{neurological headache}\right)=\frac{\text{Number of side effects in neurological headache}}{\text{Total number of side effects}}$

$\begin{array}{c}P\left(\text{Receiving drug A}\right)=\frac{\text{34+55+72}}{\text{566}}\\ =0.285\end{array}$

The formula for probability of randomly selected person with placebo and neurological headache is,

$P\left(\text{placebo and neurological headache}\right)=\frac{\left(\begin{array}{l}\text{Number of side effects in placebo }\\ \text{and neurological headache}\end{array}\right)}{\text{Total number of side effects}}$

$\begin{array}{c}P\left(\text{placebo and neurological headache}\right)=\frac{\text{34}}{\text{566}}\\ =0.060\end{array}$

Addition Rule for non-mutually exclusive events:

The formula for probability of getting placebo group or neurological headache is,

$P\left(\text{placebo or neurological headache}\right)=\left[\begin{array}{l}P\left(placebo\right)+P\left(neuro\log icalheadache\right)\\ -P\left(placebo\text{ and }neuro\log icalheadache\right)\end{array}\right]$

Substitute 0.339 for ‘ $P\left(placebo\right)$’, 0.285 for ‘ $P\left(neuro\log icalheadache\right)$’ and 0.06 for $P\left(placebo\text{ and }neuro\log icalheadache\right)$.

$\begin{array}{c}P\left(\text{placebo or neurological headache}\right)=0.339+0.285-0.06\\ =0.564\end{array}$

Thus, the probability of randomly selected person was receiving a placebo or experienced a neurological headache is 0.564.

To determine

To obtain: The probability of randomly selected person was not receiving a placebo or experienced a sinus headache.

Answer to Problem 1AC

The probability of randomly selected person was not receiving a placebo or experienced a sinus headache is 0.68.

Explanation of Solution

Given info:

The table shows the number of side effects in placebo group is 192, the number side effects in placebo and sinus headache is 11.

Calculation:

The probability of randomly selected person with no placebo is $0.661\left(=1-0.339\right)$.

The formula for probability of randomly selected person with placebo and sinus headache is,

$P\left(\text{placebo and sinus headache}\right)=\frac{\left(\begin{array}{l}\text{Number of side effects in placebo }\\ \text{and sinus headache}\end{array}\right)}{\text{Total number of side effects}}$

$\begin{array}{c}P\left(\text{placebo and sinus headache}\right)=\frac{\text{11}}{\text{566}}\\ =0.019\end{array}$

Addition Rule for non-mutually exclusive events:

The formula for probability of getting placebo group or sinus headache is,

$P\left(\text{non placebo or sinus headache}\right)=P\left(nonplacebo\right)+P\left(placebo\text{ and sinus }headache\right)$

Substitute 0.339 for ‘ $P\left(placebo\right)$’ and 0.019 for $P\left(placebo\text{ and sinus }headache\right)$.

$\begin{array}{c}P\left(\text{non placebo or sinus headache}\right)=0.661+0.019\\ =0.68\end{array}$

Thus, the probability of randomly selected person was not receiving a placebo or experienced a sinus headache is0.68.