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A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 295 people over the age of 55, 69 dream in black and white, and among 290 people under the age of 25, 11 dream in black and white. Use a 0.01 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. Complete parts (a) through (c) below.

a. State the hypotheses.

[tex]\[
\begin{aligned}
H_0: & \quad p_1 \leq p_2 \\
H_1: & \quad p_1 \ \textgreater \ p_2
\end{aligned}
\][/tex]

b. Identify the test statistic.

[tex]\[
z = 6.90
\][/tex]
(Round to two decimal places as needed.)

c. Identify the [tex]$P$[/tex]-value.

[tex]\[
P\text{-value} = 0.000
\][/tex]
(Round to three decimal places as needed.)

d. What is the conclusion based on the hypothesis test?

The [tex]$P$[/tex]-value is less than the significance level of [tex]$\alpha=0.01$[/tex], so we reject the null hypothesis. There is sufficient evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25.

Sagot :

GinnyAnswer
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.
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