Sure! Let's work through problem 10.19 with a significance level of \( \alpha = 0.05 \). Here's how we can approach this:

### Problem 10.19 Breakdown:

- **Claim**: More than 40% of all mice infected with a specific virus will die within a year.
- **Sample Data**:
  - Sample size \( n = 64 \)
  - Sample proportion \( \hat{p} = 0.48 \)
  - Standard deviation \( s = 0.05 \)
- **Hypotheses**:
  - Null Hypothesis \( H_0: p = 0.40 \) (the true proportion of mice that die is 40%)
  - Alternative Hypothesis \( H_a: p > 0.40 \) (the proportion is greater than 40%)

### Step 1: Calculate the test statistic

The test statistic for a proportion is given by the formula:

\[
z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}}
\]

where:
- \( \hat{p} \) is the sample proportion,
- \( p_0 \) is the hypothesized population proportion (0.40),
- \( \sigma_{\hat{p}} \) is the standard error of the proportion, calculated as:

\[
\sigma_{\hat{p}} = \sqrt{\frac{p_0 (1 - p_0)}{n}}
\]

### Step 2: Find the critical value and calculate the p-value

The critical value for a right-tailed test at \( \alpha = 0.05 \) can be found using the standard normal distribution.

Let me calculate this step-by-step for you.import scipy.stats as stats
import math

# Given values
n = 64
p_hat = 0.48
p_0 = 0.40
alpha = 0.05

# Standard error of the proportion
sigma_p_hat = math.sqrt((p_0 * (1 - p_0)) / n)

# Test statistic (z)
z = (p_hat - p_0) / sigma_p_hat

# Critical value for a right-tailed test at significance level of 0.05
z_critical = stats.norm.ppf(1 - alpha)

# p-value for the test statistic
p_value = 1 - stats.norm.cdf(z)

z, z_critical, p_value### Results:

- **Test Statistic \( z \)**: 1.31
- **Critical Value \( z_{critical} \)**: 1.645 (for \( \alpha = 0.05 \))
- **P-value**: 0.096

### Conclusion:

- The test statistic \( z = 1.31 \) is less than the critical value \( z_{critical} = 1.645 \), meaning that we **fail to reject** the null hypothesis \( H_0 \).
- The p-value \( 0.096 \) is greater than the significance level \( \alpha = 0.05 \), further indicating that we **do not have sufficient evidence** to reject the null hypothesis.

### Interpretation:
At the \( \alpha = 0.05 \) significance level, there is not enough evidence to conclude that more than 40% of mice infected with the virus will die within a year.

Would you like further details or explanations? 

Here are some related questions for practice:
1. How would the results change if the significance level was \( \alpha = 0.01 \)?
2. What if the sample size was increased to 100? How would that affect the results?
3. How do you interpret a two-tailed test in this context?
4. What assumptions are made in hypothesis testing for proportions?
5. How would the interpretation change if the p-value was exactly equal to the significance level?

### Tip:
Always check the sample size and the validity of assumptions (like normality) when performing hypothesis tests on proportions, especially for small samples.

Explore hypothesis testing for proportions with an example problem and solution. Calculate test statistics, critical values, and interpret results at a significance level of 0.05. Suitable for advanced undergraduate students and researchers.

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