Answer :
- Calculate the number of people still exercising in each group: $x_1 = 100 \cdot 0.58 = 58$ and $x_2 = 100 \cdot 0.18 = 18$.
- Calculate the pooled proportion: $\hat{p} = \frac{58 + 18}{100 + 100} = 0.38$.
- Identify the given values: $\hat{p}_c = 0.58$, $\hat{p}_A = 0.18$, $n_1 = 100$, $n_2 = 100$.
- The final pooled proportion is: $\boxed{0.38}$.
### Explanation
1. Understand the problem and provided data
We are given the following information:
* The proportion of adults who exercised with others and were still exercising after one year is $\hat{p}_c = 0.58$.
* The proportion of adults who exercised alone and were still exercising after one year is $\hat{p}_A = 0.18$.
* The number of adults who exercised with others is $n_1 = 100$.
* The number of adults who exercised alone is $n_2 = 100$.
We need to find the values for $\hat{p}_c$, $\hat{p}_A$, $n_1$, $n_2$, and $\hat{p}$, where $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$, $x_1$ is the number of people in the first group who were still exercising and $x_2$ is the number of people in the second group who were still exercising.
2. Calculate the number of people still exercising in each group
First, we need to find the number of people in each group who were still exercising after one year.
For the group that exercised with others:
$$x_1 = n_1 \cdot \hat{p}_c = 100 \cdot 0.58 = 58$$
For the group that exercised alone:
$$x_2 = n_2 \cdot \hat{p}_A = 100 \cdot 0.18 = 18$$
3. Calculate the pooled proportion
Now, we can calculate the pooled proportion $\hat{p}$:
$$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{58 + 18}{100 + 100} = \frac{76}{200} = 0.38$$
4. State the final answer
So, we have:
* $\hat{p}_c = 0.58$
* $\hat{p}_A = 0.18$
* $n_1 = 100$
* $n_2 = 100$
* $\hat{p} = 0.38$
Therefore, the values are:
$\hat{p}_c=58 \text{ out of } 100=0.58$
$\hat{p}_A=18 \text{ out of } 100=0.18$
$\begin{array}{l}
n_1=100 \\
n_2=100\quad
\end{array}$
$\hat{p}=\frac{x_1+x_2}{n_1+n_2}=0.38$
### Examples
This type of two-proportion z-test is often used in medical research to compare the effectiveness of two different treatments. For example, a researcher might want to compare the success rates of a new drug versus a placebo in treating a particular condition. By calculating the proportions of patients who respond positively to each treatment, they can determine whether the new drug is significantly more effective than the placebo. This helps in making informed decisions about which treatments to use in clinical practice.
- Calculate the pooled proportion: $\hat{p} = \frac{58 + 18}{100 + 100} = 0.38$.
- Identify the given values: $\hat{p}_c = 0.58$, $\hat{p}_A = 0.18$, $n_1 = 100$, $n_2 = 100$.
- The final pooled proportion is: $\boxed{0.38}$.
### Explanation
1. Understand the problem and provided data
We are given the following information:
* The proportion of adults who exercised with others and were still exercising after one year is $\hat{p}_c = 0.58$.
* The proportion of adults who exercised alone and were still exercising after one year is $\hat{p}_A = 0.18$.
* The number of adults who exercised with others is $n_1 = 100$.
* The number of adults who exercised alone is $n_2 = 100$.
We need to find the values for $\hat{p}_c$, $\hat{p}_A$, $n_1$, $n_2$, and $\hat{p}$, where $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$, $x_1$ is the number of people in the first group who were still exercising and $x_2$ is the number of people in the second group who were still exercising.
2. Calculate the number of people still exercising in each group
First, we need to find the number of people in each group who were still exercising after one year.
For the group that exercised with others:
$$x_1 = n_1 \cdot \hat{p}_c = 100 \cdot 0.58 = 58$$
For the group that exercised alone:
$$x_2 = n_2 \cdot \hat{p}_A = 100 \cdot 0.18 = 18$$
3. Calculate the pooled proportion
Now, we can calculate the pooled proportion $\hat{p}$:
$$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{58 + 18}{100 + 100} = \frac{76}{200} = 0.38$$
4. State the final answer
So, we have:
* $\hat{p}_c = 0.58$
* $\hat{p}_A = 0.18$
* $n_1 = 100$
* $n_2 = 100$
* $\hat{p} = 0.38$
Therefore, the values are:
$\hat{p}_c=58 \text{ out of } 100=0.58$
$\hat{p}_A=18 \text{ out of } 100=0.18$
$\begin{array}{l}
n_1=100 \\
n_2=100\quad
\end{array}$
$\hat{p}=\frac{x_1+x_2}{n_1+n_2}=0.38$
### Examples
This type of two-proportion z-test is often used in medical research to compare the effectiveness of two different treatments. For example, a researcher might want to compare the success rates of a new drug versus a placebo in treating a particular condition. By calculating the proportions of patients who respond positively to each treatment, they can determine whether the new drug is significantly more effective than the placebo. This helps in making informed decisions about which treatments to use in clinical practice.
