To calculate the test statistic $t$ for this hypothesis test, we use the following formula:

$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$

where:

• $\bar{x}$ is the sample mean,
• $\mu$ is the population mean under the null hypothesis,
• $s$ is the sample standard deviation, and
• $n$ is the sample size.

Given:

• $\bar{x} = 29.6$
• $\mu = 27.88$
• $s = 3.84$
• $n = 42$

Substituting these values into the formula:

$t = \frac{29.6 - 27.88}{\frac{3.84}{\sqrt{42}}}$

Calculating the denominator first:

$\frac{3.84}{\sqrt{42}} \approx \frac{3.84}{6.48} \approx 0.5938$

Now, the $t$-value calculation:

$t = \frac{1.72}{0.5938} \approx 2.90$

Thus, the test statistic $t$ is approximately 2.90.

Would you like further details, or have any questions?

Here are some related questions:

1. How do we determine the critical value for this $t$-test?
2. What is the next step in hypothesis testing after calculating the $t$-value?
3. Why do we use the sample standard deviation in this test statistic calculation?
4. What is the importance of the sample size in this test?
5. How would results change if the sample mean was closer to 27.88?

Tip: In hypothesis testing, always check if assumptions (like normality and independence) for the test are met for more reliable results.