Question

You are conducting a study to see if the probability of catching the flu this year is significantly more than 0.33. You use a significance level of alpha= 0.10

H_0 : p = 0.33

You obtain a sample of size in which there are 62 successes.

H_1 : p > 0.33

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic = __________________

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value = __________________

Answer 1

To conduct a hypothesis test for the proportion, we can use the following formulas:

1. **Sample proportion (p̂)**:
\[
\hat{p} = \frac{x}{n}
\]
where \( x \) is the number of successes and \( n \) is the sample size.

2. **Test statistic (z)**:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]
where \( p_0 \) is the hypothesized proportion.

3. **P-value**: For a one-tailed test, the P-value is calculated from the z-score.

### Step-by-step calculation:

1. **Identify the sample size (n)**:

The sample size \( n \) is not provided in your query, so let's denote \( n \) as the total sample size.

2. **Calculate the sample proportion (p̂)**:

Given \( x = 62 \) (number of successes), the sample proportion is:
\[
\hat{p} = \frac{62}{n}
\]

3. **Specify the hypothesized proportion**:
\[
p_0 = 0.33
\]

4. **Now substituting into the test statistic formula**:
\[
z = \frac{\frac{62}{n} - 0.33}{\sqrt{\frac{0.33(1 - 0.33)}{n}}} = \frac{\frac{62}{n} - 0.33}{\sqrt{\frac{0.33 \times 0.67}{n}}}
\]

5. **We need to know sample size \( n \)**:
Since we do not know \( n \), we won't be able to calculate an exact value for the test statistic and p-value.

However, if you provide the sample size \( n \), the calculations can proceed.

6. Assuming we knew the sample size \( n \), you would further:

- Calculate the value of \( z \).
- Use the standard normal distribution to find the p-value corresponding to \( z \).

Please provide the sample size \( n \) to proceed with these calculations. Alternatively, I can show you how to calculate if you give a specific sample size.