determine the P- value for this hypothesis test: In​ randomized, double-blind clinical trials of a new​ vaccine, monkeys were randomly divided into two groups. Subjects in group 1 received the new vaccine while subjects in group 2 received a control vaccine. After the first​ dose, 125 of 665 subjects in the experimental group​ (group 1) experienced fever as a side effect. After the first​ dose, 82 of 546 of the subjects in the control group​ (group 2) experienced fever as a side effect. Does the evidence suggest that a higher proportion of subjects in group 1 experienced fever as a side effect than subjects in group 2 at the alphaequals0.05 level of​ significance?

To determine the \( P \)-value for this hypothesis test, we follow these steps:

### Step 1: Define the Hypotheses
We are testing whether the proportion of subjects in group 1 (experimental group) who experienced fever is higher than the proportion in group 2 (control group). This leads to the following hypotheses:

- Null hypothesis \( H_0 \): \( p_1 = p_2 \) (The proportions are the same.)
- Alternative hypothesis \( H_a \): \( p_1 > p_2 \) (The proportion in group 1 is greater than in group 2.)

This is a **one-tailed test**.

### Step 2: Gather the Sample Data
From the problem, we have:
- Group 1 (experimental): \( x_1 = 125 \) experienced fever out of \( n_1 = 665 \) subjects.
- Group 2 (control): \( x_2 = 82 \) experienced fever out of \( n_2 = 546 \) subjects.

The sample proportions are:
\[
\hat{p}_1 = \frac{x_1}{n_1} = \frac{125}{665} \approx 0.18797
\]
\[
\hat{p}_2 = \frac{x_2}{n_2} = \frac{82}{546} \approx 0.15018
\]

### Step 3: Compute the Test Statistic
The test statistic for comparing two proportions is calculated as:
\[
z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}
\]
where \( \hat{p} \) is the pooled proportion, given by:
\[
\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{125 + 82}{665 + 546} = \frac{207}{1211} \approx 0.171
\]

Now, calculate the standard error:
\[
SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.171(1 - 0.171) \left( \frac{1}{665} + \frac{1}{546} \right)}
\]

### Step 4: Calculate the Z-value
Substitute all the values into the formula to calculate the Z-score.

### Step 5: Find the P-value
Once the Z-score is calculated, we use the standard normal distribution to find the \( P \)-value associated with this Z-score.

Let me calculate these values now.Here are the calculations:

- Sample proportion in group 1 (experimental): \( \hat{p}_1 \approx 0.188 \)
- Sample proportion in group 2 (control): \( \hat{p}_2 \approx 0.150 \)
- Pooled proportion: \( \hat{p} \approx 0.171 \)
- Standard error: \( SE \approx 0.0217 \)
- Z-value: \( z \approx 1.738 \)
- \( P \)-value: \( P \approx 0.0411 \)

### Step 6: Decision

The \( P \)-value of approximately 0.0411 is less than the significance level \( \alpha = 0.05 \). Therefore, we **reject the null hypothesis**. This suggests that there is statistically significant evidence at the 0.05 level that a higher proportion of subjects in group 1 experienced fever as a side effect than in group 2.

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. What are the assumptions underlying the Z-test for proportions?
2. How would the conclusion change if the \( P \)-value were higher than \( \alpha = 0.05 \)?
3. What other methods could be used to test the difference in proportions besides the Z-test?
4. How does the sample size affect the power of this hypothesis test?
5. What role does the pooled proportion play in calculating the standard error?

**Tip**: In hypothesis testing, always carefully choose the significance level \( \alpha \), as it affects the likelihood of Type I errors (rejecting a true null hypothesis).

Determine the P-value for this hypothesis test: In randomized, double-blind clinical trials of a new vaccine, monkeys were randomly divided into two groups. Subjects in group 1 received the new vaccine while subjects in group 2 received a control vaccine. After the first dose, 125 of 665 subjects in the experimental group (group 1) experienced fever as a side effect. After the first dose, 82 of 546 of the subjects in the control group (group 2) experienced fever as a side effect. Does the evidence suggest that a higher proportion of subjects in group 1 experienced fever as a side effect than subjects in group 2 at the alpha = 0.05 level of significance?

This detailed solution walks through a hypothesis test comparing the proportions of fever side effects between two groups in a vaccine trial. Using a Z-test for proportions, we calculate a P-value and determine whether the proportion of fever cases is significantly higher in the experimental group (group 1) compared to the control group (group 2). Key concepts include hypothesis testing, P-values, and the Z-test formula.

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