From the image, the task seems to involve choosing the appropriate test statistic for hypothesis testing based on different sampling scenarios. Let’s break down the questions:

Given information:

• Population mean: 470
• Sample mean: 488
• Sample standard deviation: 80
• Test statistic: We need to determine the appropriate test (z or t) for the given scenarios.

Question (a):

• Sample size: 14
• Population distribution: Normally distributed
• Standard deviation: Unknown

When the sample size is small (n < 30) and the population standard deviation is unknown, use the t-statistic. This is because the t-distribution accounts for the uncertainty in estimating the population standard deviation when the sample size is small.

So, for (a), the appropriate test is the t-statistic.

Question (b):

• Sample size: 18
• Population distribution: Unknown, and we know very little about it.

In this case, the population distribution is unknown, and the sample size is again small (n = 18). Since we also do not know the population standard deviation, the t-statistic is again appropriate here because we need to rely on the sample data to estimate the variability.

So, for (b), the appropriate test is also the t-statistic.

Summary of Answers:

• (a): t-statistic
• (b): t-statistic

Would you like more details on how to calculate the t-statistic or any further explanation?

Here are 5 additional questions to expand on this concept:

1. What is the difference between a z-statistic and a t-statistic?
2. How does sample size affect the choice between z and t tests?
3. How does the Central Limit Theorem affect hypothesis testing?
4. When can we use the z-statistic even with an unknown population standard deviation?
5. How does the shape of the t-distribution compare to the normal distribution for small sample sizes?

Tip: When the sample size is greater than 30, the t-distribution approximates the z-distribution, making it easier to use the z-statistic.