You are conducting a study to see if the accuracy rate for fingerprint identification is significantly different from 0.7. You use a significance level of  a=0.01 
H0: p=0.7 
H1: p is not equal to 0.7 
      


You obtain a sample of size n=687 
 in which there are 441 successes.

What is the test statistic for this sample? 

We are conducting a hypothesis test for the population proportion. In this case, the null hypothesis \( H_0: p = 0.7 \) states that the population proportion of successes is 0.7, and the alternative hypothesis \( H_1: p \neq 0.7 \) states that the proportion is different from 0.7. We can perform a **two-tailed Z-test** for the population proportion.

### Formula for the test statistic:
The formula for the test statistic in a proportion test is:

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

Where:
- \( \hat{p} \) is the sample proportion of successes,
- \( p_0 \) is the population proportion under the null hypothesis,
- \( n \) is the sample size.

### Step 1: Calculate the sample proportion \( \hat{p} \)
The sample proportion is:

\[
\hat{p} = \frac{\text{number of successes}}{\text{sample size}} = \frac{441}{687} = 0.6415
\]

### Step 2: Plug values into the formula
We know:
- \( p_0 = 0.7 \),
- \( n = 687 \),
- \( \hat{p} = 0.6415 \).

Now calculate the test statistic:

\[
Z = \frac{0.6415 - 0.7}{\sqrt{\frac{0.7(1 - 0.7)}{687}}}
\]

First, calculate the standard error:

\[
\text{SE} = \sqrt{\frac{0.7 \times 0.3}{687}} = \sqrt{\frac{0.21}{687}} \approx 0.0176
\]

Now calculate the Z-score:

\[
Z = \frac{0.6415 - 0.7}{0.0176} = \frac{-0.0585}{0.0176} \approx -3.326
\]

### Conclusion:
The test statistic \( Z \) for this sample is approximately **-3.33**.

Would you like to know more about how to interpret this result or conduct the hypothesis test? 

Here are 5 related questions:
1. What is the p-value for this Z-score, and how do we interpret it?
2. How do we reject or fail to reject the null hypothesis using this test statistic?
3. What is a two-tailed test, and why is it relevant here?
4. How does changing the significance level \( \alpha \) affect the hypothesis test result?
5. What is the effect of increasing the sample size on the test statistic?

**Tip**: Always verify if assumptions like sample randomness or normality (when needed) are satisfied before applying hypothesis tests.

You are conducting a study to see if the accuracy rate for fingerprint identification is significantly different from 0.7. You use a significance level of α = 0.01. H0: p = 0.7, H1: p ≠ 0.7. You obtain a sample of size n = 687 in which there are 441 successes. What is the test statistic for this sample?

Learn how to perform a two-tailed Z-test for population proportion in a fingerprint identification study. With a sample size of 687 and 441 successes, we calculate the test statistic using the given significance level of 0.01. This detailed guide is ideal for statistics students at the undergraduate level.

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