The question in the image involves selecting the correct test statistic for a hypothesis test based on the sample size and population distribution, and then calculating that statistic.

(a) The sample has size 18, and it is from a normally distributed population with an unknown standard deviation.

For small samples (typically $n < 30$) from a normally distributed population with an unknown population standard deviation, we use the t-statistic.

• Formula for $t$-statistic: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$ where:
  • $\bar{x}$ is the sample mean
  • $\mu$ is the population mean under the null hypothesis
  • $s$ is the sample standard deviation
  • $n$ is the sample size

So, for part (a), the correct test statistic to use is t.

(b) The sample has size 10, and it is from a normally distributed population with a known standard deviation of 3.3.

When the sample size is small and the population standard deviation is known, we use the z-statistic.

• Formula for $z$-statistic: $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$ where:
  • $\bar{x}$ is the sample mean
  • $\mu$ is the population mean under the null hypothesis
  • $\sigma$ is the known population standard deviation
  • $n$ is the sample size

For part (b), the correct test statistic to use is z.

Would you like me to calculate these statistics for you?

Here are five questions that expand on this topic:

1. What are the key differences between the t-distribution and the z-distribution?
2. How does the sample size affect the choice of the test statistic?
3. What assumptions are required for using the t-test?
4. Why is it important to know whether the population standard deviation is known or unknown?
5. Can you explain the concept of degrees of freedom in the context of the t-test?

Tip:

Always check if the population standard deviation is known or unknown when choosing between the z-test and t-test for small sample sizes.