You are conducting a study to see if the proportion of voters who prefer Candidate A is significantly less than 0.63. You use a significance level of 0.02. H0: p >= 0.63. H1: p< 0.63. you obtain a sample size of n=485 in which there are 296 successes. What is the test statistic for this sample? (Report answer accurate to three decimal places.)

To calculate the test statistic for this hypothesis test, we use the following formula for the Z-test for a proportion:

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

Where:
- \(\hat{p}\) is the sample proportion
- \(p_0\) is the hypothesized proportion under the null hypothesis
- \(n\) is the sample size

Given:
- \(p_0 = 0.63\)
- \(n = 485\)
- Number of successes = 296

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

\[
\hat{p} = \frac{296}{485} \approx 0.6103
\]

Next, calculate the standard error (SE):

\[
SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.63(1 - 0.63)}{485}} \approx \sqrt{\frac{0.63 \times 0.37}{485}} \approx \sqrt{\frac{0.2331}{485}} \approx \sqrt{0.0004804} \approx 0.02191
\]

Finally, calculate the Z-test statistic:

\[
Z = \frac{\hat{p} - p_0}{SE} = \frac{0.6103 - 0.63}{0.02191} \approx \frac{-0.0197}{0.02191} \approx -0.899
\]

So, the test statistic for this sample is approximately \(-0.899\). 

Would you like details or have any questions? Here are some related questions you might consider:

1. How do you interpret the test statistic in the context of this hypothesis test?
2. What is the critical value for a one-tailed test at a 0.02 significance level?
3. What would be the p-value associated with this test statistic?
4. How does the sample size affect the test statistic?
5. What conclusions can be drawn if the p-value is less than 0.02?

**Tip:** When working with hypothesis tests, always ensure you clearly understand the null and alternative hypotheses to interpret the results correctly.

Learn how to calculate the test statistic for a hypothesis test with proportions. Use the Z-test formula for proportions to determine the test statistic given a sample proportion, hypothesized proportion, and sample size. Example provided with detailed steps and calculations.

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