Part 1

Q1: Which test is correct to use, if we do not know the population standard deviation?

When the population standard deviation is unknown, the correct test to use is a one-sample t-test. This test compares the sample mean to a known or hypothesized population mean, with the standard deviation estimated from the sample itself.

Q2: The test statistic is -2.04. This means that the difference between the sample mean and the possible population mean, as stated in the null hypothesis (i.e., 38), is:

A test statistic of -2.04 indicates that the sample mean is 2.04 standard errors below the hypothesized population mean (38). In other words, the sample mean is significantly less than 38, but the exact conclusion depends on the p-value.

Q3: The p-value of the test is 0.043. What is the decision of the test?

With a p-value of 0.043, which is less than the typical significance level of 0.05, we reject the null hypothesis. This means there is enough evidence to conclude that the average number of polling quizzes answered is not equal to 38.

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Part 2

Q4: Which test is correct to use, if we know the population standard deviation?

When the population standard deviation is known, the correct test to use is a two-sample z-test (assuming we're comparing two groups: those attending classes and those live-streaming).

Q5: What is the null hypothesis?

The null hypothesis for this two-sample test would be: $H_0: \mu_{\text{class}} = \mu_{\text{live-stream}}$ This means that there is no difference in the average number of polling quizzes answered by students attending classes and those live-streaming.

Q6: The p-value of the test is 0.109. What is the decision of the test?

With a p-value of 0.109, which is greater than the common significance level of 0.05, we fail to reject the null hypothesis. This means there is insufficient evidence to conclude that there is a difference in the average number of quizzes answered between students attending in-person and those live-streaming.

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Additional Questions for Exploration:

1. What assumptions are made when conducting a t-test?
2. How does the p-value change when increasing the sample size in such tests?
3. What does it mean if the p-value is exactly 0.05?
4. How is the test statistic calculated in a two-sample z-test?
5. What are Type I and Type II errors in hypothesis testing?

Tip: Always check whether the assumptions of normality and equal variances hold when performing t-tests or z-tests, as this affects the validity of your conclusions.