An endocrinologist is interested in the effects of depression on the thyroid. It is believed that healthy subjects have a mean thyroxin (a hormone related to thyroid function) level of 7.0 micrograms/100 ml and a standard deviation of 1.6 micrograms/100 ml. The endocrinologist wants to assess whether the mean thyroxin level is different for those with depression. She samples 35 subjects with depression and obtains a sample mean of 7.82 micrograms/100 ml for thyroxin. The value of the test statistic for testing H0: μ = 7 is __________. (Give your answer using two decimal places)

To calculate the test statistic for testing \( H_0: \mu = 7 \), we use the formula for the \( z \)-test statistic:

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \( \bar{x} \) = sample mean = \( 7.82 \)
- \( \mu_0 \) = hypothesized population mean = \( 7.0 \)
- \( \sigma \) = population standard deviation = \( 1.6 \)
- \( n \) = sample size = \( 35 \)

### Step 1: Compute the standard error of the mean
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{1.6}{\sqrt{35}}
\]

\[
SE = \frac{1.6}{5.916} \approx 0.27
\]

### Step 2: Compute the \( z \)-statistic
\[
z = \frac{\bar{x} - \mu_0}{SE} = \frac{7.82 - 7.0}{0.27}
\]

\[
z \approx \frac{0.82}{0.27} \approx 3.04
\]

### Final Answer:
The value of the test statistic is **3.04**.

Would you like further explanation or help with hypothesis testing? 

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Here are 5 related questions you might find interesting:
1. What is the critical value for a two-tailed test at the 0.05 significance level?
2. How would the results change if the sample size were smaller, say 20 subjects?
3. What assumptions are made when using the \( z \)-test in this context?
4. How would you interpret a p-value obtained for this test statistic?
5. What alternative test would you use if the population standard deviation (\( \sigma \)) were unknown?

**Tip:** Always check if the sample size is large enough to justify using the normal approximation for the sampling distribution.

Explore how to calculate the test statistic for a Z-test when assessing the difference in mean thyroxin levels for subjects with depression. This problem involves hypothesis testing, using a sample mean of 7.82, a population mean of 7.0, and a standard deviation of 1.6. Suitable for statistics and psychology students.

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