Sleep apnea is a condition in which the sufferers stop breathing momentarily while they are asleep. This condition results in lack of sleep and extreme fatigue during waking hours. A current estimate is that 18.4
 million out of the 312.7
 million Americans suffer from sleep apnea, or approximately 5.9%
. A safety commission is concerned about the percentage of commercial truck drivers who suffer from sleep apnea. They do not have any reason to believe that it would be higher or lower than the population’s percentage. To test the claim that the percentage of commercial truck drivers who suffer from sleep apnea is not 5.9%
, a simple random sample of 337
 commercial truck drivers is examined by a medical expert, who concludes that 11
 suffer from sleep apnea. Does this evidence support the claim that the percentage of commercial truck drivers who suffer from sleep apnea is not 5.9%
? Use a 0.02
 level of significance.
Step 2 of 3 :  Compute the value of the test statistic. Round your answer to two decimal places.

To determine whether the percentage of commercial truck drivers who suffer from sleep apnea is different from the general population's percentage of 5.9%, we'll conduct a hypothesis test for a proportion. Here's how to calculate the test statistic step-by-step.

### Step 1: Define the hypotheses
- **Null hypothesis (\( H_0 \))**: The proportion of commercial truck drivers with sleep apnea is equal to 5.9%. \( p = 0.059 \).
- **Alternative hypothesis (\( H_1 \))**: The proportion of commercial truck drivers with sleep apnea is not equal to 5.9%. \( p \neq 0.059 \).

### Step 2: Calculate the test statistic
The test statistic for a proportion is given by:

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

Where:
- \( \hat{p} \) is the sample proportion.
- \( p_0 \) is the hypothesized population proportion.
- \( n \) is the sample size.

Let's substitute the values:
- \( \hat{p} = \frac{11}{337} \)
- \( p_0 = 0.059 \)
- \( n = 337 \)

First, calculate the sample proportion \( \hat{p} \):

\[
\hat{p} = \frac{11}{337} \approx 0.0326
\]

Now, calculate the standard error (SE):

\[
\text{SE} = \sqrt{\frac{0.059 \times (1 - 0.059)}{337}} \approx \sqrt{\frac{0.059 \times 0.941}{337}} \approx \sqrt{\frac{0.055519}{337}} \approx \sqrt{0.0001648} \approx 0.01283
\]

Finally, calculate the z-value:

\[
z = \frac{0.0326 - 0.059}{0.01283} \approx \frac{-0.0264}{0.01283} \approx -2.06
\]

### Final Answer:
The value of the test statistic is approximately **-2.06**.

Would you like to proceed with the next steps in the hypothesis test, or do you have any questions about the calculation? 

---

1. What is the p-value corresponding to the test statistic?
2. How do we interpret the test statistic in the context of this problem?
3. What are the next steps after finding the test statistic?
4. What assumptions must be met for this hypothesis test to be valid?
5. How does changing the significance level affect the outcome of the test?

**Tip**: Always ensure the sample size is large enough to satisfy the normality assumption when conducting a z-test for proportions.

Explore a hypothesis test for proportions focusing on the prevalence of sleep apnea among commercial truck drivers compared to the general population. Includes calculation of the test statistic and interpretation. Data includes 337 truck drivers sampled, with 11 found to have sleep apnea.

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