Sure, let's break down the steps needed to solve the hypothesis testing problem.
### Step-by-Step Solution:
#### 1. Formulate the Hypotheses:
Since we are testing whether the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25, we will use the following hypotheses:
[tex]\[
H_0: p_1 \leq p_2 \quad \text{(Null Hypothesis)}
\][/tex]
[tex]\[
H_1: p_1 > p_2 \quad \text{(Alternative Hypothesis)}
\][/tex]
#### 2. Identify the Test Statistic:
The test statistic given is:
[tex]\[
z = 6.90
\][/tex]
This statistic compares the proportions of the two groups.
#### 3. Determine the P-value:
The P-value provided for this test statistic is:
[tex]\[
P\text{-value} = 0.000
\][/tex]
The P-value is the probability of obtaining a test statistic at least as extreme as the one calculated, assuming the null hypothesis is true.
#### 4. Set the Significance Level:
The significance level ([tex]\(\alpha\)[/tex]) is given as:
[tex]\[
\alpha = 0.01
\][/tex]
#### 5. Make a Decision:
To determine whether to reject the null hypothesis, compare the P-value to the significance level:
- If [tex]\( P\text{-value} < \alpha \)[/tex], reject the null hypothesis.
- If [tex]\( P\text{-value} \geq \alpha \)[/tex], do not reject the null hypothesis.
Given that:
[tex]\[
P\text{-value} (0.000) < \alpha (0.01)
\][/tex]
The P-value is less than the significance level.
#### 6. State the Conclusion:
Since the P-value is less than the significance level, we reject the null hypothesis. Therefore, we have sufficient evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than the proportion of those under 25.
So, the conclusion based on the hypothesis test is:
- The P-value is less than the significance level of [tex]\(\alpha = 0.01\)[/tex], so we reject the null hypothesis.
- There is sufficient evidence to conclude that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25.
By following these steps, we've successfully tested the given claim and found sufficient evidence to support it.
