To test the claim that the proportion of people over the age of 55 who dream in black and white is greater than the proportion of people under the age of 25, we'll first set up the null and alternative hypotheses based on the information provided.

Step 1: Set up the hypotheses

Let:

• $p_1$ be the proportion of people over the age of 55 who dream in black and white.
• $p_2$ be the proportion of people under the age of 25 who dream in black and white.

We want to test whether the proportion of people over 55 dreaming in black and white is greater than that of those under 25.

Null Hypothesis $(H_0)$:

• The null hypothesis claims there is no difference or that the proportion of people over 55 dreaming in black and white is less than or equal to the proportion of people under 25. $H_0: p_1 \leq p_2$

Alternative Hypothesis $(H_a)$:

• The alternative hypothesis claims that the proportion of people over 55 dreaming in black and white is greater than the proportion of people under 25. $H_a: p_1 > p_2$

This is a one-tailed test because the alternative hypothesis involves testing whether $p_1$ is greater than $p_2$.

Let me know if you'd like to proceed with calculations for the hypothesis test, including the test statistic and the conclusion, or if you have any other questions!

Here are some related questions for you:

1. How do you calculate the sample proportions for both groups?
2. What is the formula for the test statistic in a two-sample proportion test?
3. How do you determine the critical value for a one-tailed test at the 0.05 significance level?
4. What is the interpretation of the p-value in a hypothesis test?
5. How do we conclude whether to reject or fail to reject the null hypothesis?

Tip: When setting up hypotheses, always make sure the alternative hypothesis directly reflects the claim being tested.