To calculate the test statistic $z$, we use the formula:

$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

Where:

• $\bar{x} = 52.9$ (sample mean)
• $\mu = 53.1$ (hypothesized population mean)
• $\sigma = 1.6$ (population standard deviation)
• $n = 200$ (sample size)

Step-by-step calculation:

1. Calculate the standard error (SE):

$SE = \frac{\sigma}{\sqrt{n}} = \frac{1.6}{\sqrt{200}} = \frac{1.6}{14.1421} \approx 0.1131$

2. Calculate the test statistic $z$:

$z = \frac{\bar{x} - \mu}{SE} = \frac{52.9 - 53.1}{0.1131} = \frac{-0.2}{0.1131} \approx -1.77$

Final Answer:

The test statistic $z$ is approximately:

$z = -1.77$

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Would you like further explanation or assistance with the next step?

Related Questions:

1. What is the next step in hypothesis testing after calculating the test statistic?
2. How do you determine the critical value for this test at a 0.05 significance level?
3. How does the test statistic relate to the p-value in hypothesis testing?
4. What does a negative $z$-value imply in the context of this problem?
5. How would the results change if the standard deviation or sample size were different?

Tip:

When performing hypothesis testing, ensure you clearly distinguish between the null hypothesis and the alternative hypothesis to guide your interpretation of the test results.