Math / Data & Statistics

QuestionIn a large clinical trial, 398,546 children were randomly assigned to two groups. The treatment group consisted of 197,285 children given a vaccine for a certain disease, and 28 of those children developed the disease. The other 201,261 children were given a placebo, and 98 of those children developed the disease. Consider the vaccine treatment group to be the first sample. Identify the values of $n_{1}, \hat{p}_{1}, \hat{q}_{1}, n_{2}, \hat{p}_{2}, \hat{q}_{2}, \bar{p}$ , and $\bar{q}$ . $n_{1}=\square$

Studdy Solution

STEP 1

What is this asking? We need to calculate some important values from a clinical trial that tested a vaccine, like how many kids were in each group and what percentage got sick. Watch out! Don't mix up the groups – keep the vaccine group and the placebo group separate!
Also, remember that $\hat{q}$ is just $1 - \hat{p}$ .

STEP 2

1. Define the groups and their sizes.
2. Calculate $\hat{p}$ and $\hat{q}$ for each group.
3. Calculate $\bar{p}$ and $\bar{q}$ .

STEP 3

The first group ( $n_1$ ) is the treatment group (the kids who got the vaccine).
There are 197,285 children in this group, so $n_1 = 197,285$ .

STEP 4

The second group ( $n_2$ ) is the placebo group (the kids who didn't get the vaccine).
There are 201,261 children in this group, so $n_2 = 201,261$ .

STEP 5

$\hat{p}_1$ is the proportion of kids in the vaccine group who got the disease. 28 kids in the vaccine group got sick, out of a total of 197,285.
So, $\hat{p}_1 = \frac{28}{197,285} \approx 0.000142$ .

STEP 6

$\hat{q}_1$ is the proportion of kids in the vaccine group who didn't get the disease.
Since $\hat{q}_1$ is just $1 - \hat{p}_1$ , we have $\hat{q}_1 = 1 - 0.000142 = 0.999858$ .

STEP 7

$\hat{p}_2$ is the proportion of kids in the placebo group who got the disease. 98 kids in the placebo group got sick, out of a total of 201,261.
So, $\hat{p}_2 = \frac{98}{201,261} \approx 0.000487$ .

STEP 8

$\hat{q}_2$ is the proportion of kids in the placebo group who didn't get the disease.
Since $\hat{q}_2$ is just $1 - \hat{p}_2$ , we have $\hat{q}_2 = 1 - 0.000487 = 0.999513$ .

STEP 9

$\bar{p}$ is the proportion of all the kids in the study who got the disease, regardless of whether they got the vaccine or the placebo.
There were $28 + 98 = 126$ kids who got sick, out of a total of $197,285 + 201,261 = 398,546$ kids.
So, $\bar{p} = \frac{126}{398,546} \approx 0.000316$ .

STEP 10

$\bar{q}$ is the proportion of all the kids who didn't get the disease.
Since $\bar{q}$ is just $1 - \bar{p}$ , we have $\bar{q} = 1 - 0.000316 = 0.999684$ .

STEP 11

$n_1 = 197,285$ , $\hat{p}_1 \approx 0.000142$ , $\hat{q}_1 \approx 0.999858$ , $n_2 = 201,261$ , $\hat{p}_2 \approx 0.000487$ , $\hat{q}_2 \approx 0.999513$ , $\bar{p} \approx 0.000316$ , and $\bar{q} \approx 0.999684$ .

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Studdy Solution

STEP 1

STEP 2

1. Define the groups and their sizes.
2. Calculate $\hat{p}$ and $\hat{q}$ for each group.
3. Calculate $\bar{p}$ and $\bar{q}$ .

STEP 3

The first group ( $n_1$ ) is the treatment group (the kids who got the vaccine).
There are 197,285 children in this group, so $n_1 = 197,285$ .

STEP 4

The second group ( $n_2$ ) is the placebo group (the kids who didn't get the vaccine).
There are 201,261 children in this group, so $n_2 = 201,261$ .

STEP 5

$\hat{p}_1$ is the proportion of kids in the vaccine group who got the disease. 28 kids in the vaccine group got sick, out of a total of 197,285.
So, $\hat{p}_1 = \frac{28}{197,285} \approx 0.000142$ .

STEP 6

$\hat{q}_1$ is the proportion of kids in the vaccine group who didn't get the disease.
Since $\hat{q}_1$ is just $1 - \hat{p}_1$ , we have $\hat{q}_1 = 1 - 0.000142 = 0.999858$ .

STEP 7

$\hat{p}_2$ is the proportion of kids in the placebo group who got the disease. 98 kids in the placebo group got sick, out of a total of 201,261.
So, $\hat{p}_2 = \frac{98}{201,261} \approx 0.000487$ .

STEP 8

$\hat{q}_2$ is the proportion of kids in the placebo group who didn't get the disease.
Since $\hat{q}_2$ is just $1 - \hat{p}_2$ , we have $\hat{q}_2 = 1 - 0.000487 = 0.999513$ .

STEP 9

$\bar{p}$ is the proportion of all the kids in the study who got the disease, regardless of whether they got the vaccine or the placebo.
There were $28 + 98 = 126$ kids who got sick, out of a total of $197,285 + 201,261 = 398,546$ kids.
So, $\bar{p} = \frac{126}{398,546} \approx 0.000316$ .

STEP 10

$\bar{q}$ is the proportion of all the kids who didn't get the disease.
Since $\bar{q}$ is just $1 - \bar{p}$ , we have $\bar{q} = 1 - 0.000316 = 0.999684$ .

STEP 11

$n_1 = 197,285$ , $\hat{p}_1 \approx 0.000142$ , $\hat{q}_1 \approx 0.999858$ , $n_2 = 201,261$ , $\hat{p}_2 \approx 0.000487$ , $\hat{q}_2 \approx 0.999513$ , $\bar{p} \approx 0.000316$ , and $\bar{q} \approx 0.999684$ .

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Studdy Solution

STEP 1

STEP 2

1. Define the groups and their sizes.2. Calculate p^\hat{p}p^​ and q^\hat{q}q^​ for each group.3. Calculate pˉ\bar{p}pˉ​ and qˉ\bar{q}qˉ​.

STEP 3

The **first group** (n1n_1n1​) is the treatment group (the kids who got the vaccine).There are **197,285** children in this group, so n1=197,285n_1 = 197,285n1​=197,285.

STEP 4

The **second group** (n2n_2n2​) is the placebo group (the kids who didn't get the vaccine).There are **201,261** children in this group, so n2=201,261n_2 = 201,261n2​=201,261.

STEP 5

p^1\hat{p}_1p^​1​ is the proportion of kids in the **vaccine group** who got the disease. **28** kids in the vaccine group got sick, out of a total of **197,285**.So, p^1=28197,285≈0.000142\hat{p}_1 = \frac{28}{197,285} \approx 0.000142p^​1​=197,28528​≈0.000142.

STEP 6

q^1\hat{q}_1q^​1​ is the proportion of kids in the **vaccine group** who *didn't* get the disease.Since q^1\hat{q}_1q^​1​ is just 1−p^11 - \hat{p}_11−p^​1​, we have q^1=1−0.000142=0.999858\hat{q}_1 = 1 - 0.000142 = 0.999858q^​1​=1−0.000142=0.999858.

STEP 7

p^2\hat{p}_2p^​2​ is the proportion of kids in the **placebo group** who got the disease. **98** kids in the placebo group got sick, out of a total of **201,261**.So, p^2=98201,261≈0.000487\hat{p}_2 = \frac{98}{201,261} \approx 0.000487p^​2​=201,26198​≈0.000487.

STEP 8

q^2\hat{q}_2q^​2​ is the proportion of kids in the **placebo group** who *didn't* get the disease.Since q^2\hat{q}_2q^​2​ is just 1−p^21 - \hat{p}_21−p^​2​, we have q^2=1−0.000487=0.999513\hat{q}_2 = 1 - 0.000487 = 0.999513q^​2​=1−0.000487=0.999513.

STEP 9

STEP 10

qˉ\bar{q}qˉ​ is the proportion of *all* the kids who *didn't* get the disease.Since qˉ\bar{q}qˉ​ is just 1−pˉ1 - \bar{p}1−pˉ​, we have qˉ=1−0.000316=0.999684\bar{q} = 1 - 0.000316 = 0.999684qˉ​=1−0.000316=0.999684.

STEP 11

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1. Define the groups and their sizes.
2. Calculate $\hat{p}$ and $\hat{q}$ for each group.
3. Calculate $\bar{p}$ and $\bar{q}$ .

The first group ( $n_1$ ) is the treatment group (the kids who got the vaccine).
There are 197,285 children in this group, so $n_1 = 197,285$ .

The second group ( $n_2$ ) is the placebo group (the kids who didn't get the vaccine).
There are 201,261 children in this group, so $n_2 = 201,261$ .

$\hat{p}_1$ is the proportion of kids in the vaccine group who got the disease. 28 kids in the vaccine group got sick, out of a total of 197,285.
So, $\hat{p}_1 = \frac{28}{197,285} \approx 0.000142$ .

$\hat{q}_1$ is the proportion of kids in the vaccine group who didn't get the disease.
Since $\hat{q}_1$ is just $1 - \hat{p}_1$ , we have $\hat{q}_1 = 1 - 0.000142 = 0.999858$ .

$\hat{p}_2$ is the proportion of kids in the placebo group who got the disease. 98 kids in the placebo group got sick, out of a total of 201,261.
So, $\hat{p}_2 = \frac{98}{201,261} \approx 0.000487$ .

$\hat{q}_2$ is the proportion of kids in the placebo group who didn't get the disease.
Since $\hat{q}_2$ is just $1 - \hat{p}_2$ , we have $\hat{q}_2 = 1 - 0.000487 = 0.999513$ .

$\bar{q}$ is the proportion of all the kids who didn't get the disease.
Since $\bar{q}$ is just $1 - \bar{p}$ , we have $\bar{q} = 1 - 0.000316 = 0.999684$ .