The National Academy of Science reported that 37%
 of research in mathematics is published by US authors. The mathematics chairperson of a prestigious university wishes to test the claim that this percentage is no longer 37%
. He has no indication of whether the percentage has increased or decreased since that time. He surveys a simple random sample of 297
 recent articles published by reputable mathematics research journals and finds that 94
 of these articles have US authors. Does this evidence support the mathematics chairperson’s claim that the percentage is no longer 37%
? 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 if the percentage of mathematics research published by US authors has changed from 37%, 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 mathematics research published by US authors is 37%. \( p = 0.37 \).
- **Alternative hypothesis (\( H_1 \))**: The proportion of mathematics research published by US authors is not 37%. \( p \neq 0.37 \).

### Step 2: Calculate the test statistic
The test statistic for a proportion is calculated using the 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.

Given:
- \( \hat{p} = \frac{94}{297} \)
- \( p_0 = 0.37 \)
- \( n = 297 \)

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

\[
\hat{p} = \frac{94}{297} \approx 0.3162
\]

Next, calculate the standard error (SE):

\[
\text{SE} = \sqrt{\frac{0.37 \times (1 - 0.37)}{297}} = \sqrt{\frac{0.37 \times 0.63}{297}} \approx \sqrt{\frac{0.2331}{297}} \approx \sqrt{0.000785} \approx 0.02802
\]

Now, calculate the z-value:

\[
z = \frac{0.3162 - 0.37}{0.02802} \approx \frac{-0.0538}{0.02802} \approx -1.92
\]

### Final Answer:
The value of the test statistic is approximately **-1.92**.

Would you like to continue with the next steps in the hypothesis test or need clarification on any part of the calculation?

---

1. How do we determine the p-value from the test statistic?
2. What are the critical values for the 0.02 level of significance?
3. What does the sign of the test statistic indicate?
4. What conclusion can be drawn if the test statistic falls in the rejection region?
5. How does the choice of significance level affect the hypothesis test outcome?

**Tip**: In hypothesis testing, the alternative hypothesis being two-tailed means we’re checking for any significant difference, not just an increase or decrease.

Explore a detailed hypothesis test for the proportion of mathematics research published by US authors. Calculate the z-test statistic using a sample of 297 recent articles from reputable journals. Discover if the evidence supports a claim that the percentage is no longer 37% at a 0.02 level of significance.

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