For the questions on this quiz assume sample 1 has 32 people with mean 11.1 and standard deviation 2.0 and sample 2 has 28 people with mean 14.9 and standard deviation 4.3.

We want to test the claim that the population that makes up sample 1 has a smaller mean than the population that makes up sample 2 with a 0.01 significance level. find the null and alternate hypothesis

To test the claim that the population mean of sample 1 is smaller than that of sample 2, we'll set up the null and alternative hypotheses as follows:

### Null Hypothesis (\(H_0\)):
\[
H_0: \mu_1 \geq \mu_2
\]
This states that the mean of population 1 (\(\mu_1\)) is greater than or equal to the mean of population 2 (\(\mu_2\)).

### Alternative Hypothesis (\(H_1\)):
\[
H_1: \mu_1 < \mu_2
\]
This states that the mean of population 1 (\(\mu_1\)) is less than the mean of population 2 (\(\mu_2\)).

### Explanation:
- **\(H_0: \mu_1 \geq \mu_2\)** implies that there is no evidence to support that the population mean of sample 1 is smaller.
- **\(H_1: \mu_1 < \mu_2\)** is the hypothesis we want to test; it asserts that the population mean of sample 1 is indeed smaller.

Since the significance level (\(\alpha\)) is 0.01, we will use this to determine the critical value and make a decision on whether to reject the null hypothesis.

Would you like to proceed with the calculations or have any other questions?

Here are some related questions you might consider:

1. How do you calculate the test statistic for this hypothesis test?
2. What is the critical value for a one-tailed test at a 0.01 significance level?
3. How does the sample size affect the outcome of a hypothesis test?
4. What assumptions must be met to perform a two-sample t-test?
5. How do you interpret the p-value in this context?
6. What are the potential errors in hypothesis testing, and how do they apply here?
7. How would the results change if the significance level were 0.05 instead of 0.01?
8. How do we determine if the variances of the two populations are equal or not?

**Tip:** Always check if the populations' variances are equal, as it affects which formula to use for the test statistic in a two-sample t-test.

Learn how to compare means of two samples using hypothesis testing with a t-test. Explore the null and alternative hypotheses, significance levels, and critical values. Example problem: comparing means of sample 1 (32 people, mean 11.1, std dev 2.0) and sample 2 (28 people, mean 14.9, std dev 4.3).

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