A survey claims that 9 out of 10 doctors recommend aspirin for patients with headaches. To test this claim against the alternative that the actual proportion of doctors who recommend aspirin is not equal to 0.90, a random sample of 100 doctors resulted in 83 who indicated that they recommend aspirin. The test statistic for the hypothesis test is?

To test the claim, we will perform a hypothesis test for a population proportion. The test statistic for such a hypothesis test can be calculated using the following formula:

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

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

### Step-by-step solution:

1. **Identify the given information:**
   - \(p_0 = 0.90\) (the hypothesized proportion of doctors recommending aspirin)
   - \(n = 100\) (the sample size)
   - \(\hat{p} = \frac{83}{100} = 0.83\) (the sample proportion of doctors recommending aspirin)

2. **Calculate the test statistic:**

   \[
   z = \frac{0.83 - 0.90}{\sqrt{\frac{0.90(1 - 0.90)}{100}}}
   \]

Now, let’s compute this step-by-step.

First, calculate the standard error:

\[
\text{Standard Error} = \sqrt{\frac{0.90 \times (1 - 0.90)}{100}} = \sqrt{\frac{0.90 \times 0.10}{100}} = \sqrt{0.009} = 0.09487
\]

Now, compute the \(z\)-value:

\[
z = \frac{0.83 - 0.90}{0.09487} = \frac{-0.07}{0.09487} \approx -0.7377
\]

### Conclusion:

The test statistic \(z \approx -0.738\).

Would you like more details or have any further questions?

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Here are 5 related questions:
1. How do you interpret the \(z\)-score in this hypothesis test?
2. What is the next step after calculating the test statistic?
3. How do you determine whether to reject or fail to reject the null hypothesis?
4. What is the significance level, and how does it affect this test?
5. Can you explain how the sample size influences the hypothesis test?

**Tip:** When conducting hypothesis tests, always check whether assumptions like normality are met, especially for small sample sizes.

Test the claim that 9 out of 10 doctors recommend aspirin for headaches using hypothesis testing for population proportion. A random sample of 100 doctors found 83 recommend aspirin. Learn how to calculate the z-statistic and interpret the results. Suitable for college-level statistics students.

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